Is it possible to give normal users the rights to tag (change and add) topics?
Yes, go to the site settings in your admin panel, then choose “Tags” on the left.
min trust to create tag and
min trust level to tag topics
Is this also possible for specific users (or groups of users)?
I don’t think so.
There are some advanced options with tag groups that might help solve the problem in another way:
You can also set some tags to staff-only using the
staff tags site setting.