At the end of the day, your signed in user will be converted into a series of claims stored in the ClaimsIdentity representing your user in HttpContext.User.Identity. You can choose to store FirstName/LastName as columns in the user table which you then can explicitly read out and convert into the appropriate claims (if desired), or you can store them directly as claims in the AspnetUserClaims table (which is just stores them as two string columns) which by default will just automatically get added to your user's claims identity. Both methods are more or less equivalent though, so its up to personal preference.
BTW the only reason you would want these in the user's ClaimsIdentity at all, is if you wanted to save a db hit just to display the name, and always use the FirstName/LastName claims in the ClaimsIdentity. If you fetch the user, and use user.FirstName instead, there isn't much value in also generating the name claims.