C major chord should be (C E G), But some play (C E G C E). How does this comply with the theoretical definition of the C major chord, which is (C E G) ? What is the concept behind playing 4,5 or 6 different notes instead of 3 exact notes for a particular chord, when playing a chord with a 6 string guitar in standard tuning.
The three notes C, E and G constitute a C major chord. Since a guitar has normally 6 strings, it makes sense to play, sometimes, all of the strings at the same time - strum. Each string can produce one of those notes, so a 6-string chord is made. If you play bottom st. 0, 5th 3, 4th 2, 3rd 0, 2nd 1 and top 0, you have one voicing of Cmaj. Probably the most used open version. You could instead play bottom 3, and/or top 3 to make 3 other voicings. They all have the magic formula of C,E and G, but in different amounts and octaves.
Imagine a 100 piece orchestra each instrument playing a C or an E or a G, in their own range. There could be 9 or 10 different octaves of each note, but the chord is still Cmaj.
Often on guitar playing jazz, a player will play just a C and an E. The C is needed to show the root of the chord, and the E shows it's major. The G actually isn't needed, as that note is heard amongst the harmonics of the root. So you could effectively play a Cmaj with just two strings on your guitar.
It is the same chord. The 3 same notes are doubled. It is just another voicing of the C major chord.
When someone writes C-E-G they are referring to these notes as pitch-classes, i.e. irrespective of any particular octave position. To write "C-E-G-C-E" is superfluous because the last two notes, C and E, refer to a pitch-class space already occupied by the first two notes.
On the other hand were someone to write C4-E4-G4-C5-E5, it would be appropriate and even necessary in order to communicate which particular frequencies are to be played.
In other words: C-E-G is general, C4-E4-G4 is specific, and C-E-G-C-E is redundant.