First off, this is not Relationship Advice, this is raised by narcissists. There are no details anywhere in your post which indicate you suspect your date was raised by narcissists, or if you feel like you were raised by narcissists yourself but honestly it's still pretty judgemental to make these assumptions about someone after only a handful of dates.

If you're not even together yet, this honestly sounds pretty controlling and overbearing. You shouldn't have this much of a say over the life of someone you're not even in a relationship with yet. This person literally just sounds like they're reserved. Not arrogant, not "quite into themselves," as you put it. At most, probably just looking out for themselves because of past trust issues.