Physically speaking, is it possible for something to exist if it has absolutely no identity?

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For something to be a ‘thing’, it needs to be representable in mind. That is its only requirement. Something can be a thing without existing. Existence simply means that it can interact with other things. Your chair can interact with you when you sit on it, therefore your chair exists.

Identity, on the other hand, implies sentience. Your chair has no sentience, therefore no identity. Rather, the identity of the thing without sentience rests in the minds of sentient beings that interact with it. Your chair is a chair by virtue of you deeming it a chair.

A different sentient being may identify the chair differently. A woodworker might see your chair as spare wood, for instance. A hermit crab may see the shell you keep in your bedroom as home rather than decoration.

So yes, things can exist without identities of their own.