Is A Sentient Weapon A Creature?
In most cases, that is what the question boils down to as most spells will clearly specify that a spell affects a creature or an object (or both, perhaps with different rules.) In some cases (e.g., Charm Person) it affects a specific subset of creatures, namely, humanoids.
So, is a Sentient Weapon a Creature?
Prrrobably not, but I don't think the answer is clear any more.
Evidence against is this Jeremy Crawford tweet, which is quite emphatic in statement, specificity, and reasoning. However, Jeremy Crawford tweets are no longer considered canon unless included in the Sage Advice Compendium. And (although it is possible I missed it on a ctrl-F fueled skim) I do not see a definition of creature or any reference to sentient items in the 2020 Compendium.
Evidence for is primarily the description of sentient items themselves (emphasis mine, below):
Some Magic Items possess sentience and personality. Such an item might be possessed, haunted by the spirit of a previous owner, or self--aware thanks to the magic used to create it. In any case, the item behaves like a character, complete with personality Quirks, Ideals, bonds, and sometimes flaws. A sentient item might be a cherished ally to its wielder or a continual thorn in the side....Sentient Magic Items function as NPCs under the GM’s control. ... A sentient magic item has Intelligence, Wisdom, and Charisma scores.
None of those words comes right out and says "This is a creature," but all of the emphasized text strongly imply some attributes of creature-ness to me. If an item is possessed, it seems reasonable to me that the possessing spirit would be a creature (specifically, an undead.) The partial stat block of Int, Wis, and Cha puts in place the machinery for creature saving throws of some types. And telling me to treat it like an NPC certainly implies to me that it ought to be treated like a creature to the extent logically possible.
Two additional questions that bear directly on this line of inquiry are: What is The Definition of Creature, and Is there anything that is simultaneously a creature and an object?
A Judgment Call
In the absence of a hard ruling from the Compendium (either that I missed, or in a subsequent update) or something similar, I think it's reasonable to rule either way. But in either case, careful readings of spell descriptions are required, and may give some odd results either way.
Identify -- This probably doesn't give a saving throw regardless the rule, because the spell admits no saving throw.
Phantasmal Force -- This targets a creature and requires an Intelligence save. Are sentient weapons (which generally have vision and hearing) meant to be immune to this? It probably wouldn't be game breaking, but it probably isn't the intent.
Disintegrate -- On the other hand, Disintegrate affects both Creatures (which get a Dex save, which a sentient weapon doesn't even have) or Objects... but not Magic Items
On balance, to the extent that we can infer the design intentions of the game, I don't think sentient weapons were intended to be creatures. But nor do I really think that was fully thought out in advance, despite later answers. I would be tempted to rule on a case by case basis where spells affecting those stats-- in reasonable situations-- might allow saving throws. In the absence of a Compendium or similar ruling, I don't think anyone could tell me I'm just plain wrong. But the same obtains for someone making the opposite ruling-- no one can say they're just plain wrong.