The same thing happened to and Shopify didn’t do anything except make me jump hoops.
At first they tell me to contact partner support, then partner support told me to submit my request through their DMCA link, then the legal team said they couldn’t find my original content in their listing, even though they even copied white spaces and grammatical errors.
TLDR Shopify is 99% likely to play blind. Depend on your app revenue and your competitor size, you should probably just hire a lawyer and submit a formal cease and desist letter.
In my case I got lucky because my competitor app is just bad, and their existence makes my niche less attractive to new comers. The one losing in this situation is actually the Shopify app store.
Here’s my related post: