Consider the following propositions:
"I am me" "I am my Father's son"
In both these cases, the predicate is the same as the subject by definition of the very subject and predicate.
Is there a special name for these kinds of propositions?
In technical logic, a predicate is an entire statement. In your usage, with respect to grammar, the predicate is the verb and object (or other parts) which apply to the subject; the predicate is usually some relation about the subject. I think the latter concept of 'predicate' is what you are referring to.
In your statements, the subject is 'I' and the predicate is 'am me' or ' am my father's son'. The predicate relation, in both instances, is an equivalence relation ('is', equals'). The object of that particular predicate is proposed as equivalent to the subject. As such a proposition is then referred to as an equivalence.
The first one, because the pronouns refer to the same thing, is a tautology, because x=x is already an axiom of equivalence relations. The second is not, because you could be female.
No i don't think there is a name. But as it stands you have NOT given an example of a sentence where the predicate IS the subject:
Your example: 'I am me'
Predicate: 'am me'.
The subject is not the same as the predicate because the subject contains one letter and the predicate four. Remember from grammar that 'subject' and 'predicate' are syntactic categories!
If you want a genuine example of a sentence who's subject is the same as the predicate then try these: a man a man, the dangerous dog the dangerous dog. is purple on the outside is purple on the outside...etc.. You just get ungrammatical nonsense.
Here is a sentence who's subject refers to the same thing as the predicate:
'The property of being red is red'.
Subject: 'The property of being red'.
Predicate: 'is red'.
Both of these refer to the property of being red. But like i said, i doubt these kinds of sentences have their own name.
Note also that identity sentences (the one's you use in your example) are not even ones where the subject merely refers to the same thing as the predicate:
The S refers to the thing a whereas the P refers to the property of being identical to a. These are not the same things.