It's possible that there could be an entirely different charge mechanism that works exactly like ours but they don't interact at all; but the whole premise of dark matter is that it doesn't seem to have any self-interaction outside of gravity.
By itself, that does not exclude the possibility of dark matter having other interactions which do not interfere with our detectors.
Since where we see dark matter mass shadows we don't see structure formation, what self interaction it may have must be very, very limited.
Don’t ask me how or why, but this is essentially the universal (pun intended) consensus amongst cosmologists.