How can I remove this strange bath vanity light?
That looks to be a built in LED light. No (easy) way to replace the light when it burns out, you are supposed to get a whole new light. This is a newish trend in lighting that I'm not in favor of. Its great for the manufactures, not for consumers or the environment.

If you are handy with electronics and know how to solder, you can replace it.

The white chip on the bottom of the large heat sink, that is the LED. Desolder it and see if there are any identification marks on it. If not  you will have to figure out what new LED chip to get from the power supply in the light. Look at the voltage and the wattage it supplies.

Assuming you can deduce the correct replacement chip, and you found a source to buy one. Then you will need to apply a thermal paste to the new chip before you screw it back down to the heat sink. Last resolder the electrical connections.

Is it worth it? Sadly, probably not.
Just get a new light fixture.