So let's not assume that.

Bruh.

Therefore, defining the identity of something requires an infinite number of definitions. — Unlimiter

Kind of sounds like the Buddhist concept of dependent origination with infinite regress. No thing is fundamentally disconnected/independent from anything else in the universe in time and space. Though any useful abstraction focuses in on a certain level of simple utility and ignores what is negligible.

The purpose of my cup this morning is to hold the tea I drank. Dog forbid the bloody causal chains that produced it since the big bang drown me in bewildering infinities.

Assuming that the identity/essence of something is defined by its purpose: — Unlimiter

It seems to me that two things can have the same purpose yet still be different, so purpose doesn't define some thing's identity. Does defining your purpose define all that it means to be you? What is your purpose?

Isnt it true, that the existence of one thing, excludes the reality of it being any other thing, than it already is and therefore a infinite ammount of definitions could be inherent to every existent thing.

What you need is to find an ultimate purpose, a purpose of all that is.

And what is the purpose of all being?

That is correct. Moreover, the identity of a thing can change from one moment to another, and it is also subject to how every thinking being in the universe can interpret that identity. The First thing in the Universe either spontaneously emerged or always existed. If Purpose is also "a thing" was there no "purpose" before the First Thing and therefore no objective purpose to the universe?