How would Chompsky categorise this?
He wouldn't, because you didn't specify a grammar. You implemented one in a Turing machine (aka perl).
The implicit, underlying grammar for parsing is not in RE. It's in CFL, even in DCFL. In this case it's LL(1) (you need one character lookahead to distinguish * and ** tokens).
Ironically you usually think of LL(1) in terms of top down parsers, that is "all grammars that can be matched by recursive descending parsers with one token lookahead" (not an exact definition, of course), but you use a bottom up technique.
The technique is none of the standard techniques I know, which all read the input string (or stream) from left to right, while you perform multiple passes (with while( m[[()\s]] )).
The computation of the result doesn't work in CFL, of course. If you're really a tough guy you can evaluate such things in context sensitive grammars (but it's much work), and for that you need, in general, a linearly bounded Turing machine (LTM).
In the normal computation model you can't modify the input tape, and the size of a supplemental, rw-tape is the size the "linear" refers to. And since the algorithm above uses a complete copy of the input string, it also uses (at least) a LBM ;-)