So it's a quite forward question. However I cannot find anyone calling this chord a Cadd8 or C raised 7th on google.. so there must be another term
There is no other term than plain old C! As long as there's a root (C), major third (E), and P5 (G), that constitutes C. That's the bare bones of C major. Put in as many more C, E or G notes as you like, and it's still C major. It's the voicing that changes, so the sound of it, subtly, but not its name.
I don't think it could even be called a tetrad, which is a 4 note chord, as in reality, it only has 3 note (C, E, G).
When you see 'add' after a chord, it indicates an extra note - as in Cadd9, which has the usual C, E and G, but an added 9th note of D. It couldn't be 'add8' as that note, C, is already in play. When you see 'sus' after a chord, the 3rd is removed, and either a 2nd or 4th is used in its place.
The formula (1, 3, 5) = (C, E, G) indicates the ingredients for the major chord. These 3 notes played alone are a triad. As Laurence Payne indicates above, repeated notes don't get special treatment. We frequently add octaves to a chord and that doesn't change the name convention (or formula). On a guitar the E-form (full 6 string bar chord) would have (C, G, C, E, G, C). Three occurrences of the 1 and two of the 5, only one occurrence of the 3.