Battery light and odometer stay lit up when the key is removed?

Hello,

It sounds like you might have a faulty ignition switch.
The ignition switch is the electrical component that the lock cylinder attaches to.
On the switch there are several pins and connector. Over time they can wear off or get loose. When that happens many time you will have no start issue or light staying on etc.
So the switch on your vehicle is probably sticking and keeping the power to the odometer and few other component. That's why you are loosing battery charger.

https://www.2carpros.com/articles/car-battery-dead-overnight

I have attached diagrams for your reference.

Please let me know of any questions.

Thank you.