Yes, this is certainly true.
In California, I currently pay almost $3.3K/yr in property taxes for the house I have lived in for over 15 years (1200 sq ft, 3/2, 50+ years old tract home). Two of my neighbors pay almost $5K in property taxes for comparable houses, which they recently purchased. The previous owner for one of these houses was paying only $1.5K, but he owned the house for over 50 years.
So property taxes for comparable houses in my immediate vicinity range from $1.5K to $5K, depending on the purchase date. One can appropriately evaluate this information at on-line sites such as zillow.com since the purchase date is available.
Still, the $3.3K in property taxes I pay as a "long-term" homeowner for a relatively small beat-up house in California is significantly higher than what I would pay in several other states that I find desirable, including states with no state income tax. And I live in a comparatively inexpensive California town with respect to property taxes.
My property taxes increase 2%/yr. I may need to live in my house for another 45 years before there is equalization with some of these other states. I guess that is something to look forward to on my 100th birthday. But if I decide to move to a comparable house in California, my property taxes will sky rocket.
So yes, property taxes for Californian's who have owned their homes for many decades may be similar to those in several low-tax states. Californian's, such as myself, who have owned their homes for one to two decades may pay twice as much as they would in low-tax states. But new homeowners will pay triple or more. One slight caveat. Californian's who purchased their homes 2-4 years ago benefit from the low property valuations at purchase time.