Why does the Thai embassy say I cannot stay more than 60 days on a 90-day tourist visa?

A 90 day visa means that the visa is valid for entry up to 90 days after it is put into your passport. IF you do not enter within that time the visa expires and you need to get a new one before you can enter.

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The allowed period of stay is 60 days.

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This can usually be extended once for 30 days more at a cost of 1900 baht at the nearest immigration office.

You are confusing the visa validity and number of days to be permitted to stay based on that visa.

Single Entry Tourist Visa is valid (<- means duration you can use that visa to enter Thailand) for 90 days from the issued date by the embassy/consulate.

And when you are approved to enter Thailand on that SETV, you are permitted to stay for 60 days (arrival date is day 1) (<- means duration you can stay).

60 days is the maximum number of days per entry on a Tourist Visa, regardless of Single Entry or Multiple Entry.

Most embassies' websites explain this. You might want to re-read that.

BTW, tourist visa entry for 60 days is eligible for 30 days tourism extension at the local immigration office AFTER you enter Thailand for most nationalities.