The Thing on this is that you need to understand the legalities of this. And how DMCA Works. They are following the Law, as setup around the world through multiple treaties. DMCA states that if you complain about it, they are Required to take it down. However after a Set amount of time they are required by law to let them put it back up. You are Required during that Down time to deal with the legal side of things.
According to https://www.dmcaagentservice.com/dmca-compliance-procedure-for-sites-and-apps/ you have to file a law suit within 10 days, or they are allowed to republish. It literally turns into a He said, she said otherwise where both people claim to be the owner.
Now from a Fraud Perspective, if Shopify gets Complaints from people who are buying from the site, with order numbers, saying they are not getting their orders [and after what is considered a respectable amount of time to take shipping into account], they will start a Fraud investigation (as opposed to DMCA Copywrite). This will then look at getting the User Banned and seeing if they can prevent the user from using the whole service including an Auto Ban on any new account that looks like its setup by the same people.
But it comes down to legality.
This really sounds stressful!
Been through this a lot and invested money but ended up other stores are benefitting from it. I came across this UNIQUE app, and it is very cool. It detects other stores that uses my brand's images and contents.. It automatically sends a notification and you have the option to remove these by yourself. Saves time in responding to DCMA counter-notices.
Try it. they have a free trial. https://apps.shopify.com/unique-anti-theft#reviews