>He’s a great dad but she doesn’t do as much as me at all.

No, he's not. He's fun dad who does all the fun stuff, but leaves all the actual work that it takes to raise a child to you.

The petty side of me wants you to start whining back at him that he does nothing around the house, except whine. Whine about his not doing chores, not helping with the baby, not helping with the cooking. Just whine right back at him.

The rational side of me says that you need to sit down with him and have a real discussion about an equitable split of responsibilities. You mention you work full-time - does he? What does HE contribute to the household.

You seem to have a handle on the most likely reasons:

• One interpretation is that he's insecure and needs validation.
• Another is that it's a power move to keep you in line by letting you know he has options so don't step out of line, or he'll access those other options.
• He may just be egotistical and likes to brag.

Next time he brings it up, just calmly ask him "I'm not sure why you are telling me this. What am I supposed to do with that information?"

You could also go with "yeah, I get a lot of offers too" and when he gets upset, respond with "oh, I thought we were sharing."

No matter how you slice it, it's a douchy maneuver.