Property taxes are locked to the purchase price of the home in California because of a law called Proposition 13. It's a been major contributor to the housing crisis but is extremely popular with everyone who currently owns property. It lets owners benefit from appreciation of their property without any increase in their tax liability while newer (and often younger) buyers must pay taxes based on the current absurd home prices.

Yes, with rare exceptions Prop 13 means you get your property tax assessment locked in when you purchase. The major way to trigger re-assessment is to pull a permit for construction/improvements, which is why unpermitted remodels/additions are so rampant in California.