I need a password manager which can work over multiple systems
…and
… if the password manager generates its own passwords , what to do if I want to access eg. banking on a eg. friends computer ? I wont know the password.
There are several self-hosted options available out there. I’d start by researching those options to see what is the best fit for your needs.
The same thing you’d do if you set the password yourself and forgot it. You’d either look it up (if the password manager allows it), or you’d go through the “forgotten password” flow for the site in question. There’s no “magic answer” to “how do I log into a system where I don’t know the password”.
You can look at using passkeys (which are generally more secure than passwords), but passkeys also tend to be device-specific. If you use, say, a hardware token plugged into a USB port on one computer, moving it to another computer doesn’t automatically let you use it (otherwise, someone could just steal your token and use it on their machine). There are ways to do that (using the same tools that are used to share passwords between multiple devices), but that’s not necessarily a default behavior.
I do not run things like banking on a friends computer but I do sometimes run an App/website on my mobile that needs a username/password and for that I use KeePassXC on the mobile. No cloud for me, I have good backups I just copy the file or USB to the mobile or mail it as attachment.
Bitwarden is pretty good & they offer a free service for personal users or you can pay if you want more features. You can access via the web or have an add on for your browser to replace the browser password manager & import those from there if you have them. Also has mobile apps for convenient access to passwords in your friend computer scenario. Hope this might help - not sure if that’s what your looking for with your “systems”. See: Bitwarden features & pricing etc.