My trailer battery (when plugged into the car 2018 Subaru Outback) connects directly to the car battery (positive to positive and negative to negative) just like you are jump starting another car.
The Alternator keeps the car battery topped off which charges the trailer battery at the same time. This bring both batteries up to the same voltage. It brings the trailer battery up to 12.5 volts which is very charged.
The solar panels and solar controller will squeeze more electricity into the trailer battery than the car but in my opinion the trailer battery is 97% charged by the car. It does not bring it up to the 13 or 14 volt float charge (fully charged).
There is a device ( forget the name of the thing) you can add to bring the trailer battery to fully charged while driving but in my case I see no need for the thing.
I have camped and left the care connected to the trailer (no shut off switch) and the result was the trailer battery did not drain very much since the two batteries together has lots more amps available than just the trailer battery. I'm not recommending that just noting my results. Make sure you leave enough charge in the car battery to start the car! The car battery is not deep cycle and should not be used for deep cycle (according to biblical law).
I know a number of people who are not using deep cycle batteries and just use an old spare auto battery for light trailer duty who have not been smited yet at the time I wrote this.
=Cosmo