The final clause may be understood either as a participle clause or as a relative clause reduced by "whiz-deletion": that is, it represents
...
which is/wasidentified with ...
In either case it modifies the preceding noun phrase, represented by that. This that is a determiner "fused" with its deleted head, a repetition of part (in traditional grammar, it's called a "demonstrative pronoun"):
... that
part...
Restore the deletion and the fusion and you get:
The problem was that the part involved was that part which is/was identified with "musical skill."
Identified with here means approximately "declared or assumed to be the same thing as".