Why can I see a button for the possibility to flag a comment next to other users' comments but not next to my own?
Flagging posts is a privilege that is documented in the help
It is not showing the flag icon because one can't flag his/her own comments. The reason is they can be directly deleted.
Flagging own comments was already requested in the Main Meta site at least twice, but it was not well received. See:
One option is to flag comments from others (or flag the question/answer over them) click on the 'other' reason and explain to moderators what you would like to do that you can't do by yourself (for example, request an edit to it).
If you want to learn more about comments I suggest reading the following posts:
And this answer, which explains why comments are second-class citizens in the Stack Exchange network; so, despite they are important we always should be prepared to see them leaving (i.e., being deleted).