I do find identities create a weakness in people. It makes it easy to manipulate others who strongly hold on to an identity. If one does not have strong ties to identity it is much less likely that they will try to defend that sense of identity.
I think a lot of what identity does, is make people feel like they belong to a group. In a literal sense, they are selling themselves, and do "belong" to that group. But then, I'm a really weird person, that thinks very weird things.
Identification can create all sorts of problems. The kind of manipulation you're talking about is incredibly common. Proselytization depends on the tendency people have to manipulate themselves into conforming to a label once they adopt it. People seem to feel compelled to justify the labels they adopt, so they start to invest in the identity by changing their behaviors and retconning their past. That's how you get social control.
But I don't think you can avoid identification. Identification is about more than joining groups; it's also about stating facts (or attempts to state facts). Every time you make "I am a ..." statements (like the part I highlighted in your post) you're verbalizing an aspect of your identity. You see yourself as a weird person, and this self-understanding impacts your behavior. You can turn "weird person" into a defensive structure ("don't be surprised by the crazy things I say, I'm a weird person"), a guiding principle ("being weird is good, so let's explore some other alternative ways to think and act"), and a narrative structure ("I've always been a weird person").
I don't think there's any way to avoid this, nor do I think it's bad. Identifying with your role as a father can make you a better father. The problem isn't identification; the problem comes from allowing an identification to lead to harm, either to yourself or another, or to the erosion of your intellectual integrity.
I think the important thing is to be aware that identities have this impact on how we think and behave and to become conscious of the process so that we're guiding it and not acting blindly or being manipulated by others. (Like how you noticed the attempt at manipulation by the car salesman.) As long as you're aware of the process, you can disavow an identity when you notice that it's having a harmful impact.
Ofc, many people are so invested in their identities that they'd rather be corrupted by them than relinquish them.