We had a loosely similar situation in which a noncustomer posted a review about us on a niche reviews website. It wasn't too bad because the "customer" rated us 4/5 stars but it was false nonetheless, probably even malicious to knock us down the website's rankings.
I asked for the review to be removed (or at least for them to challenge it, e.g. ask for a copy of the non-existent invoice). I was told no, they're unable to remove any reviews from their website because they receive too many requests but
"if you can consider an ad campaign, we can get a review or two removed on the day the ad campaign launches so let me know if you can review a quick proposal".
All trust was lost then, I could only think the website in question may have added the false review themselves to sell us advertising

They didn't even care that the review was false, they would make any review disappear as long as we handed over the dough... crooks!
We're not a Trustpilot customer but I do feel they're a bit more legit, their website suggests they take false reviews seriously and it seems as though they would remove them if you work with them.
No idea on the legal aspect, I'd imagine it's between you and the customer/author. No doubt the effort involved to take action is not worth the cost or time in most scenarios - it's going to be much easier to grit your teeth and play ball with Trustpilot.