I just installed Catalina and now everything is working fine.
Installing Glyphs should have absolutely no effect on Adobe Illustrator. Something on your system is causing this.
I have never seen anything like this. The presence of Glyphs in your Applications folder is unrelated to AI’s font menu. Therefore I also tend to think there must be a different reason.
And what do you mean by your fonts? The ones you installed in the Adobe Fonts Folder?