When we say that x is necessarily P, are we not asserting that this is the case regardless of all contingent facts, so regardless of what x and P are? And if x was P regardless of what x and P are, are we not just asserting self identity?
This is an issue Frege dealt with. If I told Lois Lane that Clark Kent is Superman, she would be very surprised. Thing is, Clark Kent and Superman both refer to the same thing, so it seems as though I'm just asserting identity. From this perspective the phrase should be as informative as saying Superman is Superman which would not surprise Lois. Frege describes the concept of "Sense" which means the "Method of reference" used by a word or phrase. I can refer to the same object with Clark Kent and Superman, but each has a different sense. Superman refers to that really strong flying guy, Clark Kent refers to that really boring nerdy guy. The extra info offered by "Superman is Kent" is that the really strong flying guy is also the boring nerdy guy, which is what surprises Lois. This exemplifies the difference between propositions of the form A = A and A = B: A and A have the same sense, A and B have different senses. Make sense?
Consider this logically necessary statement: "If everything is red and round, then everything is red and everything is round."
This sentence makes no mention of identity. It does not claim that two things are identical. And it certainly does not claim any kind of self-identity, whatever that might mean.
So, no, not all logically necessary statements "claim self-identity."
Consider: Something(x) must have some quality(P) to indeed be that thing. : Identifying an attribute(P) of something(x).
Your question is: "Regardless of what x and P are, are we not just asserting self identity?"
To give you an answer to this question, we must only show one case of x and P which is not 'just asserting self identity' -- bugger all the logical hoop jumping.
Self identity has the property that, in it's statements, you can reverse the identified terms and still get a true proposition.
Let's use your form: "x necessarily is P".
Our x is say 'A Louise Ville Slugger', and our P is the quality 'wooden'.
This gives us: A Louise Ville Slugger is necessarily wooden.
Can we reverse the terms without getting something absurd?
Nope, so for all x's and all P's, there are some possible cases in which self identity is not being expressed; the answer to your question then being: no. In all statements were P is a quality, self identity is not being expressed.
The verb " to be " has standardly 3 meanings
identity : a = b
membership : a is a B ( that is: a is a member of the set of B's)
inclusion : A's are B's ( A is a subset of set B)
The morning star is necessarily the evening star ( for these "two" stars are in fact one and the same).
Felix ( my cat) is necessarily a mammal.
The Cat ( taken as a species) is necessarily a mammal.
So " X is necessarily Y" does not necessarily mean : " X = Y", for the " is" is not necessarily an " identity - is" ; it can also be a " membership-is" or an " inclusion -is".
As to the question of knowing whether all identity statements are logically true, I think that one could answer that
yes, from an extensional point of view
no from an intensional point of view.
Necessarily " the person who won the elections against Hilary Clinton" is " the man that built an anti-immigrant wall" : this is true if I mean by this that the person denoted by the first expression is necessarily the same as the person denoted by the second.
However, the two expressions are not logically equivalent, for they do not express the same concepts: " winning against Hilary" does not logically imply " builing a wall" ( and reciprocally). So, the " person that won against Hilary " could have not been " the person that built the wall".
Note : on the distinction between denotation ( extension) and sense ( intension), and its use in the analysis of identity statements, , you may have a look at Frege https://plato.stanford.edu/entries/frege/#FreTheSenDen