Mom is ok with me dating my girlfriend, but wants it to stay "platonic"

So a lot of drama has been happening as of late. Long story short, this summer, I came out to my dad. While I assumed he had told my mom, no one mentioned the topic again, so it was never confirmed. Then, a few days ago, my girlfriend and I got into a bit of a fight (more like I was moving too fast and she was overwhelmed), so she broke up with me. I was seriously heartbroken and for the first time in several months, seriously contemplated self-harming. My dad was out of town so I called my mom crying and she helped calm me down. This is the first time that I had personally mentioned the fact that I was bi to her, and also the first time she heard that I was dating someone (this is my first partner), because my girlfriend is in the closet and was worried my parents would tell hers if they knew.

So a few days passed, and, long story short, my girlfriend and I are back together. She became extremely worried when I told her that I told my mom, and requested that I ask my mom if she would be ok with it if I attempted to get my girlfriend back. My mom said that she wouldn't tell my girlfriends parents, nor would she keep us apart, but she wasn't going to be completely approving of our relationship because she didn't think I was emotionally strong enough (because of my complete breakdown a few days prior) and she thought I was too young (16 years old) and had too many other issues to focus on.

The next day, she told me that while it was ok for my girlfriend and I to date, she would rather it stay "platonic." As in, it's ok to call one another girlfriends as long as we don't engage in anything physical. Um...wtf? Is she basically telling me it's ok to date my girlfriend as long as I only treat her as a friend?

My reaction was basically to laugh it off and tell her to stop making things awkward, and because I really don't like lying about things (the only reason I lied about my relationship in the first place is because my girlfriend was worried about her parents finding out), I told her that while I knew we were too young for sex, it was none of her business if my girlfriend and I kissed or hugged. She kind of just shook her head and gave up because she knew fighting with me about it would be a lost cause.

Still, it worries me that she wants me to be platonic in my relationship. Is this some form of subtle homophobia? Or what else could she mean by it? It doesn't even make sense to be platonic towards someone you're in a relationship with.