I believe you’re the one overthinking it in this case.
I’m 61 and female. I’ve had 2 marriages, plus s 1 long term (6 years) relationship, and I guess maybe a couple other men I was serious enough with to differentiate them from casual dating.
Not one of them ever asked me to be their girlfriend , and I never asked them to be my boyfriend.
Actually, the subject of exclusivity never came up with any of them. That’s because it was apparent we were either in each other’s company, at our jobs or school, or in some way engaged in life where being on a date with someone else wasn’t likely. I can see where that subject could be a good thing to discuss with many couples. However, I frankly never had the energy or excess time to pursue dating even one other man when I was with someone I cared about, plus everything else that life entails.
In short, he considers you his girlfriend. Simple as that. It sounds as if he has other important things going on and you being his girlfriend is one less thing he has to worry over. It sounds like it’s a comfort to him that in his busy life you are someone he can count on. You’re a positive for him.
The question is he a comfort and a positive in your life? Is he someone you can rely on and trust? Are you happy about having him in your life? If you answer yes to these, feel free to call him your boyfriend. After 6 month of a good relationship with each other, I think that’s a given.
To an extent, I admire your asking yourself if this person is at this level in your life. It seems way to many people both male & female refer to someone as bf/gf after 3 or 4 days, or because they know each other on IG.