It increases it once.

Multi-attack abilities tend to specify.

The second time you use an attack action during your turn, you take a –5 penalty to your check. The third time you attack, and on any subsequent attacks, you take a –10 penalty to your check. Every check that has the attack trait counts toward your multiple attack penalty, including Strikes, spell attack rolls, certain skill actions like Shove, and many others.

Tail Spin is a two-action ability that allows you to make a single check, with an attack trait, that affects two enemies. Many abilities that allow you to target multiple creatures (or one creature multiple times) include a rider along the lines of "this increases your MAP X times, but not until you're done with this ability," such as Slashing Gust, Double Slice, Drifter's Wake, or Whirlwind Strike, quoting the last:

[...] Make a melee Strike against each enemy within your melee reach. Each attack counts toward your multiple attack penalty, but do not increase your penalty until you have made all your attacks.

These abilities let you make multiple attacks and increase your MAP accordingly (albeit typically after the ability has concluded).

Tail Spin is one attack.

The main distinction between Tail Spin and those abilities is how many checks you're making: in all of those abilities, you are explicitly making 2+ attacks with separate rolls. With Tail Spin, you're making one attack that targets multiple creatures:

You excel at using your tail as a weapon to upend your foes. Attempt a single Athletics check to Trip up to two adjacent creatures. If you roll a success against a target, you get a critical success against that target instead.

For a counter-example, take Piston Punch:

[...] Make a single bludgeoning or piercing dynamo Strike and compare the attack roll result to the ACs of up to two foes. [...] Roll damage only once and apply it to each creature you hit. A Piston Punch counts as two attacks for your multiple attack penalty. [...]

Like Tail Spin, Piston Punch is explicitly a single attack roll applied to two targets. Unlike Tail Spin, Piston Punch says it's explicitly counted as two attacks for MAP. While the result of Tail Spin's check is applied to multiple targets, you are ultimately making one "check with the attack trait," so it only increases MAP once.