Well, he's quite likely just good enough with kids, as he's a father - which she's translating into "treats her like his own".
He may be making an extra effort with the child to impress the new girlfriend.
I'm a thoughtful type. Definitely believe in not rushing. My boyfriend wasn't off the internet, but he was only the friend of someone I didn't know well.
We took our two 4 year olds out for the day on about our 4th "date". He treated my child the same as his; meaning - he chased both round the park.
It was several more dates (actually, it was 6 weeks) including some with kids that I told my child that I was dating this man. So - still pretty quickly. He's only stayed once with her here, but that's cos we're both RP so time is scarce. But when he stayed, it was in bed with both of us. Me in middle, everyone in pyjamas.
I know some would judge. Hell, I'd judge! Or would have done, before I did it. Fact is, there's been no revolving door situation, I know my daughter's happy, and she understands that dating is a decision making process and that he may or not be permanent. To me, the problem is if you're expecting them to call a new man daddy every three months.
This woman might be an arse - or she might be someone who has seriously thought through the situation and made decisions based on her child's personality, like me.