Q1: 'Philip sees a lot of himself in Harry= Harry and Philip have much similarity, eg. hobby, characters, between themselves? Yes, Philip recognises much of his own character and tastes in Harry.
eg1. My father see a lot of himself in me. (correct?)
eg2. I see a lot of myself in my father. (correct?)
Certainly when the children are young it would be the parent who recognised her/himself in the child, but I think that from adolescence, children start to see themselves in their parents.
Q2: Will the subject and object of the below sentence sound a bit of ambiguous? I mean two 'he' and readers have to distinguish which 'he' refer to. Should I avoid such writing? Or it is ok? In the context of your text it's fine, not at all ambiguous.