SF Bay area, I've had 3 formal house assessments, & each time they "start with" Zillow et al to look at "reasonable values in the neighborhood". Which seriously affects your ability to refi, etc. So Yeah Zillow has an effect. It's worth it to create a login & enter in your upgrades, so your ZESTIMATE goes up.
Who is doing these "formal assessments"? Are these broker price opinions (performed by a real estate broker)? It seems very odd they would use Zillow data instead of the MLS to find comps.
I did it before Zillow, and they start with the property value as estimated by the city for tax purposed and look at recent sales that are similar in the area, and adjust.