These tax savings pass to homeowner families who stay in one location for multiple decades, especially as land values rise.
Homeowners are very opposed to being displaced, since moving means their taxes would reset to a higher level.
Renters are excluded from these savings.
Not exactly. It is more complex than that. The tax base is supposed to be the market value but if the market value raises, the tax base is allowed to follow only up to a certain margin. With CA and especially Bay Area prices skyrocketing for a while, the effect has been than the tax base lags very far from the actual market value, and for people who bought in the 80s the tax base may be a fraction of their current market value.
When you buy the house your tax rate is 1% (+ local fees) of your purchase price. Every year thereafter, it goes up by at least 2%.
The assessed value will go up by 2% (that's prop13) so the base tax will go up by 2%. However, local city and county can add their own fees on top of that (so they do), therefore in many years the actual dollar amount increase will be more than 2%.
(Source: my property tax bill over the last 23 years.)
Source: just bought my first home in CA. Taxes were surprisingly reasonable
Also your property taxes start out at 1% of assessed value, which is very low compared with many other high-wealth states.