The final clause may be understood either as a participle clause or as relative clause reduced by "whiz-deletion": that is, it represents

... which is/was identified 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".

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/was identified 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".