The typical process for installing any distribution of Linux is that you download an “ISO” from a web site. You don’t have to provide any details in order to do that download. It is as anonymous as any other download. That is to say, your IP address most likely appears in a web server log somewhere and will be retained for an unknown amount of time (could be short, could be long, could be forever). Technically the same applies to any other information that your web browser sends with a download request. Most of that other information can be falsified if it is of concern to you.
If you don’t wish to download an ISO from a web site then some distributions also make the ISO available via BitTorrent. (In that case though you should check the integrity of the ISO before using it.)
Once you have that ISO, what you do with it is up to you. I am not aware of any distributions that force you to supply details in order to install but even if that happened, it is unlikely that there will be a problem if you were to supply fake details.
I think you have right that I can not use Lutris on the PureOS.
Since yesterday I have been trying to find out if I can play World of Warcraft Classic on the PureOS, but I can not find any additional information and thats pity. I hope that someone is gonna help me.
Search the forums for running wine in pureos, that’ll be the first (and probably most difficult) step. Maybe someone reading this can give answer. I’m under the impression it can’t be done, but I may very well be wrong.