Merriam-Webster
1
: to encroach upon in a way that violates law or the rights of another
infringe a patent
2
obsolete
: DEFEAT,
FRUSTRATE
I'm going to keep looking, but so far I haven't found anything that indicates a picture ID to be an infringement. But maybe setting only certain days to vote is such an infringement. Maybe having to actually put pencil to ballot or pull a lever is such an infringement. Maybe they should send someone to find me and ask me to voice my choice...no, having to actually speak might be an infringement.
You have the right to vote. An infringement is something that materially hinders your ability to exercise it. Needing to show an ID does not do that, nor does any other modest requirement designed to keep elections as clean as possible.
The same applies to guns. If showing an ID is an infringement on your right to own one, it's because you shouldn't own one.