To anyone having a similar problem: we solved it actually easily after we realized we could specify the session cookie name ourselves, per (sub)domain.
So, we adjusted the config/session.php cookie
key:
'cookie' => env('SESSION_COOKIE_NAME', 'laravel_session')
and then we set the SESSION_COOKIE_NAME
variable in respective .env files, like this:
# subdomainA.example.com
SESSION_COOKIE_NAME=subdomainA_session
or
# subdomainB.example.com
SESSION_COOKIE_NAME=subdomainB_session
etc.
Hope this helps.