How do I handle 'see other people' talk while dating?

So we’ve been seeing each other for five months, right? Things got serious fast — dinners at my mom’s, weekend trips, the whole deal. Then last night, after wine, she goes, 'I love you, but I think we should see other people. I don’t believe in possessive love.' I just... froze. Like, what even is that? Are we a couple or a suggestion? I told her it feels like she wants the comfort of us but won’t commit. She said I’m 'too intense,' that love shouldn’t feel like a cage. But that’s not the point. I’m not asking to lock her down, but if we’re sleeping together, sharing holidays, talking every day — shouldn’t that mean something? My brother said in Provence they’d call this 'un flirt qui se prend au sérieux,' like I’m taking it too seriously. But I don’t see it that way. Am I being old-fashioned? Or is she just scared? I don’t want rules for the sake of rules, but I need to know where I stand. Right now I feel like the backup plan... 💔