In logic, the 'law of identity' classically says (...) More abstractly, For all x, x=x.
The law of identity which is assumed in formal logic is not x = x but "x is identical to x". Applied to numerical values, equality is fine but misleading outside them. The Eiffel Tower is identical, not equal, to the Eiffel Tower.
In logic, the 'law of identity' classically says: 'Every thing that exists' has a specific nature.
No. Saying that everything that exists has a nature has nothing to do with the law of identity.
To say that something has a nature is to say that some aspect of it remains constant over time, and the law of identity does not say that. Things that change over time are still identical to themselves because identity is identity between the thing and itself now. A variable may take different values as long as the same value is given at the same time to all its occurrences.
What does it mean by "For all x" or 'every thing that exists' or 'every being that exists' mathematically or logically?
Mathematicians are not interested in everything. They are only interested in a particular subset of everything, namely, mathematical concepts, so they always assume some domain.
Formal logic is obviously more general than mathematics and so does not require assuming any domain. As long as you don't assume that a variable is somehow restricted, it is not, and so by default it is thought of as potentially identical to any value whatsoever.