Albertsons is aware of this issue. I spoke with an employee in their payment services department and they've been working on a fix. He tried to suggest it was an issue between American Express and Apple but I was quick to point out that my American Express works with Apple Pay just fine at EVERY other retailer I try (that accepts Apple Pay). It's only the Albertsons owned companies that have an issue with the "Not Authorized" error.
Also, for the record, Safeway is NOT the parent company of Albertsons. I retired from Albertsons 10 years ago and I'm VERY familiar with the mergers and buyouts in the supermarket and drug industries over the last 50 years.
Albertsons is privately held by Cerberus Capital Management. Cerberus acquired its first Albertsons stores in 2006. Other Albertsons stores were acquired by SUPERVALU that same year. Seven years later the SUPERVALU Albertsons were struggling and SUPERVALU sold them to Cerberus. Cerberus subsequently purchased additional chains, including Safeway/Vons.
Cerberus operates a great number of supermarkets under their respective names but all part of the Albertsons family umbrella. See:
https://en.wikipedia.org/wiki/Albertsons
https://www.albertsonscompanies.com/about-us/history.html
Look at the list of supermarket and drug companies at the bottom of that second link. If those chains are all on the same payment system software as Albertsons, Vons, Safeway, etc. then you will experience the same "Not Authorized" error message when attempting to use American Express (and certain MasterCards, apparently) with Apple Pay in those stores.
I have friends that still work for Albertsons and Vons. They explained that the major issues (with Apple Pay) started when Albertsons companies switched to using the point of sale software that was used by Safeway/Vons (prior to Cerberus acquiring Safeway/Vons). The very day that switchover happened in each store, Apple Pay stopped working entirely. It took weeks to get it working again. Later, the point of sales terminals became finicky and some would work and other wouldn't. Managers had to reboot the POS terminals in order to get Apple Pay working again.
This latest "Not Authorized" issue is the worst of all. It seems NOBODY knows why it is happening. And, to some degree, nobody seems to care.