Battery Light Stays On

There are some wiring connections that need to be confirmed. Here is a guide to help you do some testing along with a alternator wiring diagrams so help you see how the system is connected.

https://www.2carpros.com/articles/how-to-check-wiring

There is also three fuses that run the systems as well.

https://www.2carpros.com/articles/how-to-check-a-car-fuse

Check out the diagrams (Below)

Please let us know what you find. We are interested to see what it is.

Cheers, Ken

You need to go to a well known auto part store such as O'Reilly Auto Parts or AutoZone and have your battery and alternator tested for free. That is usually an indication that either your battery or alternator is failing.

You need to have your charging system checked out could be the alternator is on its way out-causing the light to come On.

Your scenario can be quite frustrating.
Alternaters starters and thermostats are the worst offenders when they are bad out of the box.

You need to check for a draw, Ideally a mult-meter is the best tool but a test light may also work if the draw isn't too significant.

If no other electrical accessory is causing the battery to drain overnight a manual draw check of the electrical system will need to be performed. What this means is you will be checking the electrical draw the battery has on it when the car is locked up, with the key in the "off" position". First open the hood and disable the under hood illumination light, if equipped. Next, with the key off and the doors locked wait 15 minutes, then disconnect the battery cable on the negative side. (The 15 minute wait allows the computers to go into "sleep mode" and shuts down all electrical). Attach a test light between the negative battery cable end and the negative battery terminal. The test light should illuminate dimly or not at all. If the test light is on brightly there is a strong electrical draw in the system. To locate this electrical draw start removing fuses one at a time. When the test light goes out the circuit in question has been located. You will need a car repair manual to identify all accessories in a particular circuit, repair as needed and re-check system. Again a multi-meter set in milli-amps is a preferred method.

IF there is not exceptional draw, then I would start looking at the alternator.

IS it really "new" or just new to you. Meaning, I expect this is a remanufactured alternator.
These can be a bear to nail down when they are failing intermittantly. I have been through the ringer with this scenario and sometimes it requires going to a truley new one or sometimes an OEM one.
If so what brand is it?

Most likely the multi-plex control system underneath the dash has failed. These are hard to diagnose because some parts are wireless and read frequencies. I'd bring it to a electrical diagnostic shop or Honda to have them confirm what the problem is.

Something is going on with the charging system.

Some common problems:

Alternator failure
Power wire that runs behind the intake manifold is rubbing through to ground.
Blown fuse.