As I stated in the title, I'm able to share my filters and dashboards with other groups in Jira, only when I'm in the group too, which is very weird. For example there is a group "company-management", and only when I add me into this group, I can see this group in the dropdown field. So that means, I see only the groups, where I'm as member in them.
Global permisssion seems however fine, and it's Jira cloud instance (paid version).
This is indeed the current behavior: you need to be a member of groups/roles in order to share filters with them. There is a related open suggestion: As an Admin I want the ability to share the visibility of a filter with Groups and Roles that I am not a member of, you can upvote and watch it to be updated if it is released.