The spell doesn't say.
But multiple lines of evidence suggest the duplicate is not solid and can be detected.
'Spells do only what they say they do' only gets you so far
"Spells do only what they say they do" is a great principle when appropriately applied. I think the best application of this principle is when limiting the power of a spell. For example, a recent question asked about using encode thoughts to gain a tool proficiency. In my answer, I pointed out that while the spell does say it can transfer "a memory", it does not say that it can transfer a tool proficiency, and thus ascribing it the ability to transfer tool proficiencies was too powerful, as it permitted the spell to do something it did not say it did.
However, "spells do what they say they do" can also be used to argue for making a spell more powerful, perhaps limitless, and this is not an appropriate use of the principle. Mislead says "You can use your action to...make [your illusory double] gesture, speak, and behave in whatever way you choose." Can I make the double speak a language I do not? Can I make it attack? Can I make it multiattack? Clearly there are some implied limits on 'behave in whatever way you choose', and we shouldn't worry that imposing such reasonable limits violates "spells do what they say they do".
Sometimes, "spells only do what they say they do" produces confusing results, often when what you would like to know about is simply not addressed in the spell description. Then your conclusion may depend on how you construct your statement (i.e. 'the spell doesn't say you can't, so you can' (permissive) vs 'the spell doesn't say you can, so you can't' (restrictive)). In the specific case of mislead, the spell does not directly say whether the illusory double is solid or can pass through things. Whichever way you rule makes the spell more powerful in one way, but less powerful in another. 'The spell doesn't say the double is solid, therefore it is not, so I can have it walk through a locked door and scout the other side.' 'The spell doesn't say the double is intangible, therefore it is not, so I can have it pick up objects.'
Given the inadequacy of the principle "spells do what they say they do" when (like mislead) the spell doesn't actually say what it does, that is, when a major part of its effects is simply undescribed, we need another methodology.
You can compare the spell to other, similar spells
Each spell is a unique entity, and we certainly need to be careful when making such comparisons. But if there exist other, better-described spells with similar functions (especially in the same school), such that we can reasonably conclude they are roughly versions of the spell in question albeit at different levels of power, then we can use the other spells to get an estimate of the unwritten parameters of our spell of interest. In the case of mislead, a number of other spells present themselves.
The functions of mislead
The fifth-level spell mislead does a number of different things, all of which are presaged in lower-level spells that do only one such thing each. Mislead makes you invisible, like the second-level illusion school spell invisibility. It makes an "illusory double" of yourself, much like the "illusory duplicates" of the second-level illusion school spell mirror image. And it gives you a movable scrying sensor, like the fourth-level divination school spell arcane eye. Since mislead combines all of these functions, it is not surprising that it is a higher level than any of them. Although mislead does not do everything these other spells do (it cannot be cast on others, or upcast for more effect, it makes only one image, it does not supply the sensor with darkvision...), in the absence of contrary description these spells can be reasonably be taken as suggesting a lower limit for mislead's power.
On the other hand, the seventh-level illusion spell project image is largely a more powerful version of mislead. True, it does not make you invisible, but then it is not designed to instantly mislead anyone as to where you are; rather it serves to project your person-duplicating scrying sensor much further than mislead can. It lasts much longer, and rather than starting in your space it can begin up to 500 miles away and go from there. In fact, apart from the invisibility, the spells are otherwise so similar in their functions that a line-by-line comparison is worthwhile.
First paragraph (your double, duration):
(Mislead) You become invisible at the same time that an illusory double of you appears where you are standing. The double lasts for the duration, but the invisibility ends if you attack or cast a spell.
(Project Image) You create an illusory copy of yourself that lasts for the duration. The copy can appear at any location within range that you have seen before, regardless of intervening obstacles. The illusion looks and sounds like you but is intangible. If the illusion takes any damage, it disappears, and the spell ends.
Second paragraph: (movement and action)
(Mislead) You can use your action to move your illusory double up to twice your speed and make it gesture, speak, and behave in whatever way you choose.
(Project Image) You can use your action to move this illusion up to twice your speed, and make it gesture, speak, and behave in whatever way you choose. It mimics your mannerisms perfectly.
Third paragraph: (senses)
(Mislead) You can see through its eyes and hear through its ears as if you were located where it is. On each of your turns as a bonus action, you can switch from using its senses to using your own, or back again. While you are using its senses, you are blinded and deafened in regard to your own surroundings.
(Project image) You can see through its eyes and hear through its ears as if you were in its space. On your turn as a bonus action, you can switch from using its senses to using your own, or back again. While you are using its senses, you are blinded and deafened in regard to your own surroundings.
Fourth Paragraph: (detection)
(Mislead) [None]
(Project image) Physical interaction with the image reveals it to be an illusion, because things can pass through it. A creature that uses its action to examine the image can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC. If a creature discerns the illusion for what it is, the creature can see through the image, and any noise it makes sounds hollow to the creature.
Based on their striking similarities, I think we can use project image as a default interpretation for any questions we have about mislead with the exception of the invisibility it grants.
Of immediate note is that while mislead simply doesn't say whether the illusory duplicate is tangible or detectable (hence the OP's question) project image quite explicitly says that it is intangible and detectable. As a lower-level spell, we would not expect that mislead would be better than project image, at least in the things that they both do. It is reasonable that if a projected image was detectable as an illusion, then a misleading duplicate would also be detectable, rather more reasonable than that there is no way to detect the misleading duplicate. It also seems more reasonable that an error in editing led to the exclusion of mislead's 'fourth paragraph' than that the spell was intended to be undetectable but didn't see fit to mention that fact.
The tangibility of mislead, however, is not as clear cut. The duplicates produced by mirror image can be disrupted by weapon attacks. Likewise, the scrying sensor of arcane eye can be restricted by solid barriers. Project image, on the other hand, is explicitly intangible, and because of this has greater value in information gathering and durability. The question then remains whether mislead is enough of an improvement over mirror image and arcane eye to be intangible, that is, to be as good as project image in this aspect, and here the simple comparison of spell level won't help. There are other lines of evidence we can follow, though.
Other Lines of Evidence
The nature of the illusion school
It is not useful to debate whether, or to what extent, illusions are 'real' - they all produce effects either in the world, or in the minds of creatures. However, is is important to know whether or not they can be acted on by physical objects, creatures, and attacks. The default for the school is that they cannot. Interestingly, project image is the only illusion spell to explicitly say that its product is intangible, but then 'intangible' is not a term commonly used in the rules. So how do illusion spells typically signal their intangibility?
In many cases, the 'detection clause' of the spell specifies that they can be recognized as illusions through physical interaction because "things pass through it", as is the case with minor illusion, disguise self, silent image, major image, hallucinatory terrain (where 'the tactile characteristics are unchanged'), seeming, programmed illusion, and project image. Notably, mislead has no 'detection clause', but as I argue above I believe that was an editorial error.
For illusion spells that create independent images, these images typically appear 'at a spot within range'. Notably, they do not have the restriction of having to appear at an unoccupied spot within range such as spells that summon creatures or sizable solid objects. The clear implication is that illusory images can appear in occupied spaces, and thus can co-occur with solid objects. See silent image, mirror image (where the space is explicitly occupied by you), Nathair's mischief (where the space explicitly may be occupied by creatures), major image, mislead itself (where the space is explicitly occupied by you), programmed illusion, and project image. Here we know at least that the illusory copy in mislead can explicitly pass through you, even if it is solid for other interactions.
Some spells do produce illusions with solid, tactile components - and many more than just creation, as Momonga-sama's answer claims. In fact, at least shadow blade, phantom steed, creation, mirage arcane, simulacrum, and illusory dragon all have tactile components. We know this, because the spells explicitly say so. It is safe to say that if illusions that are tangible they should have spell descriptions that indicate them as such.
In the absence of a statement to the contrary, then, we should assume that the misleading duplicate is both intangible and can pass through other things.
Lack of AC and hp
When a spell produces something that is tangible, we can expect other game entities to interact with it, and likely to attack it. Thus each spell should assign its creation both AC and hp, as spells like phantom steed do. Even if you assumed that the misleading duplicate was intended to be tangible but the spell neglected to say so, you still have no basis for deciding what the AC and hp of the duplicate would be.
The mislead spell simply does not say whether or not it is tangible or detectable, and thus we cannot use the 'spells do only' principle to decide. However, multiple lines of evidence allow us to confidently conclude that it is intangible but detectable.
Can it interact with the environment (e.g. open doors, pull levers)?
Not physically - it cannot move anything. But you can certainly have it react as if it could, for example, pretending to fall into a pit, straining against a door, etc.
Is it solid? Does it have weight (or can it be made to seem to have weight)?
No, no, and only to the extent that you can have it falsely appear to.
Can the illusion be revealed? How can the illusion be revealed?
Yes, probably through the same method as project image: physical examination permits an Investigation check.
