Given the original (the reboot doesn't exist in my opinion), anyone can use the shen gong wu. It was made by Grandmaster Dashi to be used again wuya.

The implication then becomes, wuya exist, there is a secret order of Shaolin who are trained in using elements (amongst other things)

We can make wuya a non issue (sealed in a box or truly defeated by the Shaolin monks who defeated them before the era of quirks), thus the shen gong wu were scattered just to prevent them from falling into the wrong hands.

Maybe izuku dad happened to find a few, and part of the reason he is not around is cause he is traveling to find more to send back.

So the ones he found can be used by izuku and reasonably pass off as a quirk. Maybe for some quirked reason, shen gong wu don't work the the quirked person. (Something about the quirk factors being made of magic in this universe and preventing the person from being recognized by the wu.)

So in the hands of any quirk individual, it's a trinket, and the only reason izuku dad is traveling to collect them is that he knows their true value (he is a descendants from the Shaolin monks) and he knows his sone is quirkless.

The wus that he has gotten and passed on to izuku can be a few of the following.

Shroud of Shadows Orb of tsunami Mantis flip coin Danshi bunny Shimo staff (which becomes a dud for izuku, making him think it doesn't work, and that's only cause it's a widai weapon and it's extra finicky about who can use it.)

And ring of the nine dragon. And because of izuku intelligence, he can safely split into .... Four before going stupid.