The width of your tyres does matter - it's possible to have too wide or too narrow a tyre for a particular rim. It's not recommended to put a narrow 23/20c tyre on a wide touring/city rim e.g. Mavic A719/Rigida Grizzly, because it'll come unseated or bottom out easily. Similarly you wouldn't want to put a wide 35c tyre on a narrow road/race rim, because it'll roll off whilst cornering, or the pressure will split the rim apart.
Having said that, a Charge Plug on its original wheelset will be fine with a 23c tyre.
The width of your tyres does matter - it's possible to have too wide or too narrow a tyre for a particular rim. It's not recommended to put a narrow 23/20c tyre on a wide touring/city rim e.g. Mavic A719/Rigida Grizzly, because it'll come unseated or bottom out easily. Similarly you wouldn't want to put a wide 35c tyre on a narrow road/race rim, because it'll roll off whilst cornering, or the pressure will split the rim apart.
Having said that, a Charge Plug on its original wheelset will be fine with a 23c tyre.