Given the nature of the business, when a stripper gets you drunk and then asks you personal information like what type of job you do and your name, is it possible she's stalking you for private information?

If they potentially know my real name and possibly where I work, anything I should do to avoid setting myself up for a disaster here.

It's entirely possible she's just making smalltalk, to create some imitation of intimacy. She may be trying to think of you as a person, not as an ATM machine. If she knows your name and where you work, do you think she is going to show up at work, in some scene reminiscent of the restaurant scene in the Blues Brothers? Do you think she is going to blackmail you by saying you were at a strip club? If you're really worried, in the future, just remember to give false information and be consistent about it if you see the dancer in the future.

You need to think about having some self-control.  We're all here for fun, but if you are the type to lose control, do something and regret it later, you can get in REAL deep here.

First of all, I stopped going to strip clubs ever since I discovered escorts.  But before then, I usually had a fake identity: name, line of work, etc.  Oftentimes, I picked something out of the ordinary and interesting, such as working for a national railroad or being a foreign exchange student.  This protected my identity, and let the stripper to make small talk about my "job".