I would argue you can't be obsessed with being normal and actually be normal. To obsess over it is to be weird in itself so given that, it will depend if this is coming from the 3rd person or not.
If you are talking about judgement from a negative 3rd person who is in fact likely the obsessive (with being not-normal, non-mainstream, 'unique') then they might use Pretentious, Sycophant, Conformist (as user1993 already suggested), Townie or Beta (both colloquial)
A more positive 3rd person may call them a Social Butterfly or Amiable
But, if this were a first-person assessment of themselves, they are more likely to say Plain, Normal or Average.
However, if this really is a person obsessed with the pursuit of normal, then they will probably see themselves as a Weird outsider (hence the obsession) and perhaps describe themselves that way or with the word Wannabe or Nobody
And in the same case, the negative 3rd person may (incorrectly) say they are an Attention Seeker (incorrect as they are trying to blend in, not stand out), Insecure or Needy
I'm basing these suggestion just on observations over years conducting interviews and working in bars, events and other industries, which is why they lean towards colloquial use and are not necessarily correct use of language