Is is possible within the settings of project templates to set created teams in the “Template access” as default?
Hi @Lukas_Gschwendtner , welcome to the forum 
Unfortunately not, but by default, upon project creation, the selected team will be the team that the template belongs to, in case that helps.
Note however that Template access (step 3 of the template editor) is different to the default ptoject members that will be added to a project that is created from a template (step 1, top right corner)
Template access is merely the people who can use and edit the template itself.