If I play the Cmaj chord on the C major scale
As Todd says in the comment, You don’t play chords “on” scales. As we said in How to figure out which scale to play a specific chord in?, when you play a chord, you don't have to know what scale to play it in.
Generally, when you're playing - you don't think what chord to play, then think about what chord you're playing. That's thinking backwards!
What you might do, though, is think the other way round: FIRST consider the scale you're playing on (or, as we'd more normally say, what KEY you're playing in), THEN consider what chord you would build on particular notes.
So you might think - "Hey, I'm in C minor - if I want to build a chord on the note C, what chord would it be? Aha, it would be the C minor chord".
Or: "Hey, I'm in C Major - if I want to build a chord on the note C, what chord would it be? Aha, it would be the C Major chord".
But you wouldn't usually think "I want to play a C Major chord... now what scale should I play it in?". That doesn't really make sense.
"Chords have different notes on different scales" is not true. (It's sometimes true that we might give the same chord different names in different keys though).
But "different scales (or keys) have different chords" is true.