In Canada, the "on him" usage is dying out in favour of "is wearing" or "is carrying" (especially among non-native speakers), but it's not at all unusual to hear someone say "Does he have his phone on him?" I've heard this usage far and wide, in both Canada and the US, and I've read it a lot in books. That being said, I've also heard the variant without him, especially in American speech.
But you know what they say: a preposition is the wrong thing to end a sentence with. :)
To this English speaker, asking if he has his "phone on him" or his "gym-clothes on him" sounds like you are asking if he is carrying them, not wearing them. Much the same as "on his person". "Does he have his clothes on", without the "him" gives me a clearly separate meaning.
Some English native speakers, especially in the US, use shirt to cover more or less anything worn on the upper body, so we have to accept that translation. In the stricter sense, shirt is only skjorta though. (Those people may call that 'a dress shirt' or 'a button up').
Also, Duo's policy against profanity is that we shouldn't use it at all here, not even with stars or abbreviations, so please edit your comment if you can.
Thanks as always Arnauti! In french we kind of have that to, it's more likely to happen in a familliar setting, you would sometime hear the word Haut used to talk about any clothing you could wear on your torso. (genre: J'aime bien ton haut; c'est sympas comme haut; J'ai vu un haut trop bizarre l'autre jour...)
Anyway I will avoid those mistakes in the future. ^^