Because of Jake Grossman's comment, I eventually realized the $ in the regex trigger should be escaped:
(Also had to use python interpolation)
snippet "\$(.+)\$" "whatever" r
`!p snip.rv = match.group(1)`
The trigger right after snippet will match anything between $ and capture it in a group (with the ())
In the ...
So the specific issue you've been having is caused by Vimwiki taking over the Markdown file type globally. You have a few options to prevent Vimwiki from doing that, which should help you solve your issue.
First, you could set the g:vimwiki_global_ext variable to 0, disabling it from hooking into files anywhere with those specific extensions. You can do so ...