I have debugged this and the problem is located in GiteaSCMSource.
It inherits jenkins.plugins.git.AbstractGitSCMSource which as has two retrieve methods that are overridden. One of the retrieve methods are called when an OrganizationFolder is scanned (and retrieves correctly repository data from Gitea and set the sshRemote variable for re-use.
The other retrieve method is called when a Multibranch Pipeline job is executed. In this case no metadata is collected from Gitea and the sshRemote variable is not set. Hence the only way to have this set is to re-scan the OrganizationFolder.
PR created: https://github.com/jenkinsci/gitea-plugin/pull/34
Until there is a fix in the plugin, I setup an init hook script. The script is doing a rescan of all organizations after Jenkins starts up: