No, the taxi union does not set the taxi fares. SINTRA (Secretary of Infrastructure and Transport) sets the rates. The taxi union can only ask for a rate increase (which they have) but SINTRA must approve it (which they have not, since 2016).
Many things in Mexico have price controls: gasoline, electricity, tortillas, etc. The prices of these commodities and services are set for the lowest common denominator (the Mexican poor) even though tourists and the Mexican upper-classes can also buy the same commodities or services at that low, government-controlled price.
However, there is a problem created when tourists overpay when the taxi drivers charge them more than the published rate (which is still very high for a poor family who needs the service to carry old people to the hospital or doctor, go for groceries, take the family to church when it is raining, etc.) The problem is that it makes the taxi drivers loath to work for the published rates for the folks that live in the poorer colonias. Long ago, this wasn’t a problem, nowadays, taxi drivers would rather wait in line at the piers or malecon to get a chance to pick up a tourist and make double, triple, or even ten times the published fare than they would if they made a 26 or 27 peso trip in town for a resident. This situation got so bad, that SINTRA now makes the taxi union dedicate a small (very small) portion of the taxis to take a daily rotation serving only in town so the residents can at least have some kind of service, even if sporadic and hard to find.
The SINTRA car is a red car with SINTRA printed on it. It is constantly pulling taxis out of line because it was supposed to be that taxi’s day in town, but the drivers would rather ditch the town beat and go for the over-paying tourist. It just gets worse and worse every year, because so many tourists take the attitude “It’s only a seventy-cents or a dollar more, I can afford it and the taxi driver is poor.” OK, maybe it will make you feel good like you are giving to a charity when you over-tip and over-pay a taxi driver, or knowingly let him overcharge you, but it is really hurting a lot of other poor people you don’t see. You are helping deprive them of the taxi service they need.