If it were me, I would break it down to her as if she's a child and let her know the dinners she's going on are called dates, it is not okay to go on dates with guys she met on dating sites and if she continues to do so, you will walk away from this.
This is the best advice in this thread.
There's lots of bad advice in this thread (one post that's about 210% off course). First, realize there is no 100% definitive right or wrong solution to this.
A number of moving parts here:
Is it fishy? Yes.
Is it disrespectful. Maybe, leaning yes.
Do you own her? No
Can she have friends? Yes
Do people at first go on a date and find out later they're better off platonic? Yes, but not often do they reconnect under the "platonic-only" guise.
Do people meet people again whom they initially met online to gauge interest again for romantic reasons? Usually, yes.
If it were me, in a very confident no BS tone, I would talk with her and open with something like:
"Sally, I've been thinking about your "dates" (and they are, don't let her use her conniving, "Oh, he's just a friend" BS that you'll almost certainly get from her--BIGGEST cop out in the female playbook) with the guys you previously met on dating sites. I'm a bit confused as to why you feel compelled to meet them, again, for what appears to be another date when you and I are dating exclusively. It may be ok with some guys, but it's not ok with me. So feel free to go out on the dates and have a good time, but if that's your choice, I don't think we should be exclusive any longer. I'd still like to see you, but just know, like you, I'll be seeing other people too."
Then just go silent and DO NOT CAVE OR NEGOTIATE. She may cry, call you silly, jealous, etc. Just smile and say you mean what you said. And then DON'T LINGER. Get off the phone or walk out. Don't do this at your place so she can linger. YOU want the option of literally WALKING AWAY.
Nothing could be a more powerful weapon than your WILLINGNESS TO WALK.
Let us know how it goes.
PS- A girl I was dating a number of years ago pulled the same crap with a guy at work she was incessantly talking about and having what I call an
emotional affair with. I went over to her place unannounced, said my piece and walked out. His name came up 95% less and her IL went up. That was within our first year of dating. We ended up staying together for 9 years and a lot of it was due to the standards I set early on. I was tested on many occasions and usually used the "Some guys might be ok with it but I'm not" angle. It always worked.