Question! From a New Player ! Help !

I downloaded this App for my 10YO son. He took a chess class before Covid and was taught how to play. I never was taught to play chess and sort of randomly have played through my life. I don’t even know what most the pieces do & I’m wild. I say this because it’s important to my question that everyone knows I literally know next to nothing.

My son gets extremely upset playing this game. He says it’s broken or glitched. He said that when he has my king in check, I should only be able to move the king. Several times I’ve been able to move adjacent pieces to kill the piece that has my king in check.

Can someone explain what he means? And why it occurs here?

I postulated to him that maybe there are different sets of rules depending where you are from? Some that would allow certain play and some that might not?

But I have no idea. Please someone take pity on a tired mom and explain this to me.

"different sets of rules depending where you are from"
No there are the FIDE Laws of Chess valid in the whole world
https://www.fide.com/FIDE/handbook/LawsOfChess.pdf 
3.9 The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces,
even if such pieces are constrained from moving to that square because they would then
leave or place their own king in check. No piece can be moved that will either expose the
king of the same colour to check or leave that king in check.

There are 3 was to react to a check:
1) moving the king to a square where he is not in check
2) capturing the piece that gives the check
3) putting a piece between the king and the piece that gives the check if it is not a knight that gives check