Ah, right, because of google.com and google.co.uk and such. The search engines I use, don't have the money for so many domain names, so it's just a UI selection. 🫠
With the address bar rework, you could register both of them as a search engine and then simply select the other one to change language/region.

Copying the URL could be problematic, yeah. I don't have the update yet, so I wouldn't know, if it's something real simple like just right-clicking the address bar. If you really do need it, then it should still be possible to browse to the search engine's webpage directly and issue your query via the search box on the webpage.