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

> ... <s>which is/was</s> 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 <s>part</s> ...

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".