Does the hour meter on the motor keep reading hours if the ignition is on but the motor is not running?

It shouldn't that would totally defeat the purpose of the "engine hour meter"

I thought any time key was in run position hour meter ran whether engine was running or not.

It depends, the old style 80's, 90's would run when you turned the key.

It depends, but yes, that it typically the case.

Why the question?

I was hooking up a stereo on a single battery boat and thought it needed to be wired into ignition instead of direct to battery where I know someone will forget to turn it off.

You would need a 4 way ignition switch for that. Accessory / Off / Run / Start

Psst: 99% of boaters would kill a battery that way too. Doh !

Yes...The hour meter will run, but the hours on the computer only go up if the engine is running. Don't ask me how I know... LOL

Put the REM/ignition wire on a rocker switch.

Put a battery kill switch in the boat, shut it off when you call it a day.