Does Apple Watch track padel?

The hardware does, the built in Workout app does not have a padel type. Apple Watch measures heart rate whenever you wear it, but a padel match only lands in Apple Health as a workout if something starts a workout session. Either you start a generic workout by hand, or a padel app starts one for you when the match starts.
What the watch records on its own
Worn on the wrist, Apple Watch samples heart rate through the day and contributes movement to your activity rings. What it does not do without a workout session is attribute any of that to a padel match, so there is nothing in Fitness afterwards that says you played.
There is also no padel entry in the Workout app's list. The usual workaround is to start Other, which records duration, heart rate and an energy estimate, and shows up in Health as Other rather than as padel.
What a padel app adds
A padel app can start the workout session at the same moment the match starts and end it when the match ends, which is the difference between a 74 minute match and a 74 minute workout you forgot to stop.
Because the app also knows the score, it can put the health numbers next to the result. Average and peak heart rate, active calories and duration sit beside the set scores and the points won, so the physical and the sporting record are one object.

Reading the heart rate trace of a match
Padel is intermittent, so the trace looks like a saw rather than a plateau. Rallies push it up, the walk to collect the ball pulls it down, and the gaps between games are visible as dips.
Averages hide that shape. A 147 average with a 178 peak over 74 minutes describes a very different match from the same average with a 158 peak, which is why the graph is worth more than the two numbers.
Battery and phone independence
A workout session keeps the heart rate sensor sampling continuously, which uses more battery than idle wear. A 90 minute padel match is a small cost for a modern Apple Watch, but it is not free, and it is worth starting the match at the first serve rather than in the car park.
The session does not need the iPhone. Scoring and recording both run on the watch, and the data moves to the phone when the two are next in range.

Questions people also ask
Which Apple Health workout type does a padel match use?
There is no padel type in HealthKit, so padel apps write the session under a related racket sport or the generic other type, with the duration and active energy attached.
Do I need my iPhone with me for the workout to record?
No. The watch records the workout on its own and syncs it to the iPhone and to Apple Health afterwards.