As the title suggests: a Warlock with a pact to a genie in D&D 5e has someone cast identify on their genie's vessel.
What information would that person get?
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This GM would be conservative in what information Identify can reveal.
Specifically, identify says:
If it is a magic item or some other magic-imbued object, you learn its properties and how to use them, whether it requires attunement to use, and how many charges it has, if any. You learn whether any spells are affecting the item and what they are. If the item was created by a spell, you learn which spell created it.
The genie pact warlock's vessel says that it:
Your patron gifts you a magical vessel that grants you a measure of the genie's power. The vessel is a Tiny object, and you can use it as a spellcasting focus for your warlock spells.
While you are touching the vessel, you can use it in the following ways:
Bottled Respite. As an action, you can magically vanish and enter your vessel, which remains in the space you left. The interior of the vessel is an extradimensional space .... While inside, you can hear the area around your vessel as if you were in its space. You can remain inside the vessel [for a few hours]. Any objects left in the vessel remain there until carried out, and if the vessel is destroyed, every object stored there harmlessly appears in the unoccupied spaces closest to the vessel's former space. ...
Genie's Wrath. Once during each of your turns when you hit with an attack roll, you can deal extra damage to the target ... determined by [the patron's type].
The vessel's [not quite trivial to destroy].
(Tasha's Cauldron of Everything, p 73-74, probably with some transcription typos)
So, rolling through identify:
This GM would definitely not include either the owning warlock nor the gifting genie among the vessel's properties, any more than I would include the owner or the crafter of a +1 sword among its properties. While the owner and gifter are facts about the vessel, I would be hard-pressed to call them properties. This is doubly so with spells like Legend Lore, which exists to provide hints about the history of important items. As an answer to a possibly-related question put it: "Identify would reveal that an axe is +3, but only Legend Lore would reveal that it's The Axe of Dwarven Lords".
Frankly, letting identify reveal the owner or the creator of a magic item is simply too powerful, especially for a 1st-level spell (even one with a costly material component).
The vessel is a "magical vessel" and a "tiny object", so it qualifies as a "magic-imbued object" for Identify. Thus the caster learns its properties: basically all of the text describing it on the page.
Up to interpretation
As a DM, I'd rule that which warlock it belongs to and which genie provided it falls under its properties, as well as whether the warlock is currently residing in it and whether it has any objects stored in it.
Warlock it belongs to may also fall under "how to use it" because to use it you must be that warlock. Learning the number of charges implies you learn about its magical state, so I think learning the Genie who owns it and if the warlock is inside seems reasonable.
A DM would be well within reason to not reveal this information. If your worried about a player casting this on your lamp, talk to your DM
Other magic may intervene
However, none of this information might be available if the vessel is under a spell like Nondetect or, depending on interpretation, Arcanist's Magic Aura, which a clever DM or Warlock might use for story reasons.