Dating

If I’ve met a woman three different times and we’ve been talking basically for two months, should I have her real phone number and last name?

We didn’t meet up for a few weeks because of different things that delayed it, but we just met for the third time, I’ve bought her lunch twice, and I asked if we were at the real phone number last name stage yet and she said maybe, then was saying that she’s been hurt before so she’s taking it slow and being friendly to start, I get that but at this point I feel like it’s perfectly reasonable for me to have this information.
If I’ve met a woman three different times and we’ve been talking basically for two months, should I have her real phone number and last name?
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