Since it introduces a subordinate clause, I would call it a subordinating conjunction. The clause "as he called it" is a subordinate clause; it contains a finite verb, and is embedded in a larger subordinate clause ("who didn't understand...called it").
The more difficult part of your question is whether this is an adjective clause. The analysis of subordinate clauses varies, and grammarians don't always agree on the correct categorisation for them.
As I understand it, the term "adjective clause" is usually used (often in grammar for students of English as a foreign language) to describe subordinate clauses that modify nouns. These are predominantly relative clauses - clauses that begin with relative pronouns such as "which", "that", and so on:
The coat that I bought is too big
"that I bought" is a relative clause, acting as a modifier of the noun "coat".
Although "as he called it" doesn't start with a relative pronoun, I would say it still functions to modify the noun phrase ("Muggle money"), so you could make a good case for calling it an "adjective clause".