Owen batteries are not agm batteries but gel cells. They are usually used as backup batteries in alarm system, emergency lighting etc. they have a lifetime of about 3 years and are mostly used for a few times in emergencies. The break controller receives its main power from the car and while doing so supplies a charge to the gel battery. The charge in it is just sufficient to engage the hydraulic brakes a few times in case the trailer gets unhooked while towing and pulls the emergency cable. Better not be on a slope since it won’t hold the trailer for long.
When storing the battery can and should be removed during extreme heat and cold. For charging you would use a charger set to gel batteries chemistry. If charged to fast the gel will dry out and can not be replenished like with a flooded battery. They especially do not like gasing out and will swell when doing so. Gel batteries that are swollen have reached their end of life and should be properly discarded. They contain lead and the acid is in gel form. Advantages are that they can be mounted in any orientation similar like agm batteries
They can be charged just by applying some voltage as present in the trailer supply voltage but that rough charging is not gentle on the battery. Since it is mostly in charged state and the charged voltage is about what you get on a car there is probably little harm done. However if discharged, a proper charging regimen is better for the life of such batteries. Considering the cost of those devices, that advice is frequently disregarded