In my repo, the branches usually are specified explicitly what remote should push to (to origin or to fork).
Unfortunately lightweight tags (that I am using just for development, at the moment) don't have the same option.
I've wrongly done git push --tags and all of were copied to the wrong remote. Lot of dirt; so I had to clean up all of them manually.
How to specify what remote the tags should be pushed by default?
NOTE: I usually push lightweight tags on fork, as bookmarks for later reuse (to mark commits that I don't want into a branch).
There is no default remote for tags, only for branches. When you push a tag without specifying the remote, it will go to the default for the branch you are on, aka the remote-tracking branch.
Use git branch -vv to see the tracking branches for each local branch.
Use git branch -u <desired upstream remote> [<branchname>] to change them. The last arg is optional and defaults to the same name as the local branch.