According to Grammar Monster, a phrase does not contain a subject. I argue this is not true. Grammar Monster uses to following sentence to illustrate its point.
I stopped believing in Santa Claus when my mother took me to see him in a department store, and he asked for my autograph.
'my mother' is said to be a phrase and I agree. However, 'my mother' is also acting as a subject because 'took', which is a finite verb, attaches to its subject which is 'my mother'.
Source page here...
I stopped believing in Santa Claus when [my mother took me to see him in a department store], and he asked for my autograph.
A clause consists of a subject and a predicate, prototypically a noun phrase and a verb phrase respectively.
Phrases function as subjects, as well as having other functions like object, modifier, complement.
"Function" is a relational concept. When we say that "my mother" is subject, we are describing the relation between it and "took", or between it and the whole clause. It is the subject of the clause, not simply a subject.
So it's not possible for a subject phrase to contain its own subject, since it is the subject; "my mother" is a noun phrase functioning as subject of the clause in which it functions.