I’m not sure how your site is setup. On one of mine where I can multi-host, the second domain would end up being based off a subdomain. So file structure below root like:
index.php -> main site index.php page
system -> system folder
second_site -> is a subdomain- second_site.domain1.com which they will then point domain2 to.
Once they point domain2 to the subdomain? I can still access pages at either domain2.com or second_site.domain1.com or domain1.com/second_site/
In the above setup? Logging into the backend WOULD see me as logged in on second_site.domain1.com or domain1.com/second_site/ - and this issue would be moot unless I was going to domain2.com- in which case I’d typically use an admin.php in second_site because only files in that folder are accessible with domain2.
BUT I am not all that up on the different ways servers might be setup to handle multiple domains, so I can’t swear yours would work the same way.
Though just thinking it through- if the main site was the subdomain with the system folder in it- it could hit the system folder via domain1.com/system and then the second site could hit it domain2.com/second_site/system. But if you added another subdomain and pointed domain3 at it- couldn’t use domain3 and get to system, would need an admin.php.
And that was long! But I think it just boils down to- if the system folder was a simple file system.html- can you get to it using the subdomain url you want? If so- shoud be easy to fake /system login (because you aren’t faking it). Otherwise- not that I know of and I’d be a little leery of server magic that would make that happen. Which is not to say definitively that it’s impossible.