Possible, but multiple DM adjudication points required
So let's suppose we have James, a rogue, manacled to a table...But our hero has an ally, who has a plan. He casts Banishment on James, who fails his save and disappears from the table.
James' ally1 casts Banishment, and James fails his save2, ok.
Presumably the manacles stay behind, since they're fixed to the table and not something he was wearing.
Yes, his gear goes with him3. But is he not wearing the manacles? Unfortunately, while many spells interact specifically with 'worn or carried', the 5e rules don't actually define these terms. And in natural English, I could certainly say 'the prisoners were wearing heavy manacles'. 'Anything on your character sheet' has been suggested as a working definition of what is worn or carried. But that might interfere with your conception of how spells work and verisimilitude. Why should the Banishment spell take the metal bracelet on my wrist with me but not the metal manacle on my wrist? Or, how does a firebolt burn the rope binding me to a tree but not the rope in my backpack? Do the spells somehow know what is 'mine', or check in with me about what I consider mine?
Thus a DM decision is required here. You could have Banishment free James from the manacles. Or he could appear in the other plane with the manacles still attached. Or with the manacles and the table! Probably the upper limit would be the weight of all the objects attached to him, since he can't carry more than his carrying capacity, which is a game-defined term.
In order to proceed, let's assume that your instinct as a DM is to leave the manacles behind.
Now what's unusual is that this causes the target to be incapacitated -- can't take actions -- but does not stop them from moving...Whilst in the demiplane, James can't take actions, but he uses his movement to stand up from prone.
An incapacitated James stands up from prone4.
The spell does bring the target back right where they left, but interestingly it only says "in the space it left"...Then his ally ceases concentration and James reappears on the table, but standing up and free from his bonds, ready to fight. Is this possible? I can't see anything in the text of the spell that requires James to reappear in the exact same position he left from
One consequence of 'spells do only what they say they do' is that spells should not impose conditions without explicitly saying so. Prone is a condition. Thus, returning from Banishment should not impose the prone condition. If he was standing up in the plane-of-Banishment, he should be standing up when he returns. Now, this might be on top of the table, but as you say, the only requirement is that it is in the same space from which he left, and as a Medium creature, that is a 5' x 5' area. So it could also be crouching under the table. It could also be standing next to the table. DM decision required here, too.
Although you don't explicitly say so, presumably while he was in the manacles, he had the Restrained condition as well, but that depends on what you as a DM have decided that these specific manacles do. If you have decided that the manacles impose the restrained condition while he is in them, and that the manacles do not travel with him, then you would likely rule that while he is banished he is not restrained (just incapacitated). His return from Banishment should not impose the restrained condition, just as it should not impose the prone condition.
Suppose he did not choose to stand up from prone while in the other plane. When he returned, he would still be prone, and perhaps even still in the same position on the table, since the spell does not say it stands someone up upon their return. But even without him trying to escape, the simple act of his banishment would likely result in his return free of the manacles! Thus to free him, his ally could simply successfully cast Banishment and then immediately drop concentration on it, without having to wait long enough for James to have stood up.
while it might step slightly on the toes of freedom of movement (which is also 4th level), it's probably of limited enough utility that it's not a problem.
I would not worry about it as a balance decision. With Banishment, you have used a 4th level spell to escape the prone and presumably restrained conditions once. Freedom of Movement is a 4th level spell that provides that ability for a hour, without Concentration. No toe-stepping there.
1Note that Banishment can only target a creature "that you can see within range [60 feet]", so it is a bit strange that the ally is permitted close enough to view to overly-complicated and slow-acting death trap. But perhaps they are cleverly hidden or have just burst in, sure.
2As you say, for the plan to work, James will need to fail his save. RAW he cannot choose to fail the save; however this might be a case where you as a DM decide that the situation permits him disadvantage. You might even allow, as designers have suggested, a voluntary fail.
3Spells do only what they say they do, and Banishment only says it sends the target creature to another plane - so anything it was wearing or carrying would be left behind, right? No, actually. Unfortunately you have to look in Sage Advice to find out that unlike polymorph-type spells which explicitly specify 'worn and carried', teleport-type spells implicitly assume the target's gear goes with them.
4As you correctly state, incapacitation does not prevent movement. However, just to be completely clear, it would not matter if it did. You do not need to be able to move in order to stand up because standing up is not movement. You do need to be able to spend movement, since standing up costs half your speed in movement, so it would be more precise to say 'incapacitated does not prevent him from spending movement, so he uses his movement to stand up from prone.'