If I go out somewhere with a new GF, I take note of what she chooses to eat, if she picks the most expensive things. Similarly, if we go shopping and I offer to buy her something, whether she picks expensive things,
or whether she says that's too expensive and picks something more modestly price. In other words, does she take care of (my) money or spend it all too readily! 
I would not be too quick to tell a new GF about my financial situation, if anything I usually tell them I don't have much money and see if they stay around. But it is an interesting point, that if in due course, the subject of "you buy(me)house" crops up, it can be interesting to tell them "I have bought one already". And then wait and see the reaction.
If she is pre-occupied with whose name it is in, or whether it can be transferred into her name, rather than just pleased that she may be able to live there with you, then that would concern me.
I was once looking at apartments to buy, with a GF at that time. She was very interested and helpful, until she found out that it would be sold on a lease and would be in my name. Then she didn't think it was a good idea at all!
