Make an interweaving quine
Fission, 6 bytes

'!+OR"


Try it online! Try two copies! Try three!

Explanation

This is just the Fission standard quine. It happens to work for this challenge, because Fission has explicit entry points into the program. In particular, by duplicating the program, we add another R which adds another atom (instruction pointer). Since the source code is toroidal, the effective code being executed doesn't change otherwise — to each atom, the code still looks the same locally. However, the atoms are executed in lock step, so that the things they print are interleaved and we get an additional copy of each character in the output.

For completeness' sake I'll just shortly reiterate how the program itself works. Regardless of whether we repeat the program or not (e.g. '!+OR"'!+OR"'!+OR"), each atom sees the following code:

R"'!+OR"'!+O


The " toggles string printing mode, so that the program starts by printing '!+OR directly to STDOUT, which is all of the quine except the quote. Then '! sets the atom's mass to the character code of !, + increments it, which gives ", and O prints it while simultaneously destroying the atom. The program then terminates, because there are no atoms left.