Ok, so it definitly happens on more curious (intrusive) websites.
Due to the lacking extension support at this stage I mostly use Vivaldi Mobile for sites that I like and trust when on the go. At home I block Ads using Pi-Hole.
So out of curiosity I just went to welt . de, a news website belongig to a German publishing house with great power and little morals and the rather large amount of cookies and trackers you would expect from a site like that, and what do you know? The Google Play Services Notification appears.
If it helps I can give you a query log from my Pi-Hole, showing which domains are blocked when visiting sites like that. I will do some experimenting over the next couple of days and try to gather some reliably documentation on when exactly this happens.
But as I said, it happens on other Browsers as well, so it might just be a Chromium or even Android (which you can probalby never completely de-Google...) related issue.
Btw: Nice to see the chart about Vivaldis popularity on the appbrain link you provided. Seems to be a good start đŸ™‚