The vessel cannot be the target of a summons spell
Answer
The vessel itself is a tiny object on its outside. However, the
interior of the vessel is an extradimensional space in the shape of a 20-foot-radius cylinder, 20 feet high, and resembles your vessel.
The instant summons spell works only when:
You touch an object weighing 10 pounds or less whose longest dimension is 6 feet or less.
The "longest dimension" of the vessel is its interior diameter, which is 40 feet across, far larger than the 6 feet permitted by instant summons. Thus, you cannot cast instant summons on the vessel.
Commentary
In his answer, Jack considers several possible interpretations of what might happen, and added mine after I wrote my answer. For full disclosure, I think the RAW are vague enough to justify other interpretations besides the one I describe here. However, this interpretation recommends itself to me because of its practicality - a feature I greatly value as a DM. If there are multiple possible interpretations of the rules, and one lends itself to permitting shenanigans and abuse while another does not, I will generally come down with the more restrictive one (a principle Jack himself describes in his answer to Are you considered out of range of spells and weapons when inside the hole of a Portable Hole?).
To respond to Jack's specific objections to the interpretation that the vessel is too big to have an instant summons cast on it:
"a) it presumes instant summons can somehow know the vessel's extra-dimensional interior dimensions"
While there are spells that do have to 'know' things (such as magic mouth and glyph of warding), instant summons doesn't work like that. Rather, it can only be cast on a certain class of objects. Trying to cast it on an invalid target means the spell simply fails - the same as casting a sacred flame on a statue you thought was a gargoyle or trying to use raw materials that you did not know were magical for fabricate. The caster does not have to know that the vessel is bigger on the inside, and the spell does not have to know that either - the spell simply fails because a legitimate target was not chosen as part of casting it.
"b) this interpretation limits instant summons in regards to a variety of items such as portable holes"
I consider this a feature, not a bug. A portable hole is already a powerful enough item, and I'm okay with it not being able to be summoned remotely or retrieved after theft. As above, being able to summon your hole may allow shenanigans or trivialize challenges.
Not being permitted to use summons on your hole doesn't substantially diminish the power or utility of the hole or the summons spell, just their interactions.
"c) it questions whether instant summons would work on a coiled up rope or rolled up rug."
Yes, it does do that. But instant summons is only one of six spells in the PHB that can be cast on objects, but which are limited to objects of a certain size (the others being light, mending, nondetection, stone shape, and wish). Light is a cantrip. Insisting that instant summons works on a genie vessel because you don't want to deal with complicated definitions of dimensionality is not going to help you with things that are far more likely to come up, like casting light on a coiled rope.
In his comments, Matthieu says "This answer assumes that the vessel's extradimensional space is part of the item itself. Since all other instances of such extradimensional space items do not "contain" their extradimensional space (merely acting as an access point, or portal), I believe this assumption needs some form of support or other proof for it to hold."
Yes, I do assume that the extradimensional space is part of the vessel, largely because the description says "The interior of the vessel is an extradimensional space" (and not just that the vessel 'connects to' an extradimensional space, like the description of the efficient quiver says). The space is the interior of the vessel; the interior of the vessel is part of the vessel. If a caster inside the vessel could not cast instant summons on the room that was forty feet across, than they cannot cast it on the same vessel when they are outside it, even though the outside part of the vessel is tiny.
As far as the assertion that no other extradimensional item works like this, I agree that the mouth of a portable hole does act like a portal, and the inside of the hole is on another plane. However, that space is still part of the same item, and the hole does indeed 'contain' the space, just as the description says: "The cylindrical space within the hole..."
The description of an efficient quiver doesn't give us much to be able to tell either way, but by saying the compartments 'connect to' an extradimensional space, I am comfortable accepting that the space is not part of the compartment or quiver itself.
On the other hand, a bag of holding expels all its contents if it is turned inside out. If the mouth of the bag is just a portal to another space and the physical bag's interior is not that space, why should turning the bag inside out affect anything in the space? Also, piecing the bag from the inside, if possible, shouldn't rupture the physical bag if the space and the bag are not the same thing. A handy haversack has just these same "inside out" and "piercing from within" clues that the extradimensional space is physically the interior of the item itself.
