'22 1250 GSA next week. '17 RTW ready to trade.
With my '05 RT (hex-head with canbus), I would charge the battery through the aux port, but you had to turn the key on, then turn the key off, then plug in the charger while dash lights were still on. It's in the Rider's Manual. It would keep Can-bus working until battery fully charged, then it shut off and wouldn't charge again until you repeated this process...so... yeah, it works through the aux plug way, but no, you can't maintain the battery for a whole winter season that way unless you come back and turn the bike on every few weeks.
I'm not a believer in leaving a charger plugged in all the time anyway. I always charge until full , then repeat every few weeks. The glass mat batteries hold a charge for a long time without going down if you don't have a drain on them.
I don't think the new bikes are any different in the way the Aux plugs work. My '22 GSA came with a dealer-installed, direct to battery SAE plug that hangs down by right foot peg. It has a little rubber cover. I just unplug the cover and plug in there to charge as the dealer did when I need to. My dealer puts them on any bike that sits in their showroom.
BTW: The OP was asking about the charger in the photo. It will work great with AGM batteries. You can buy the same charger in a different case from an aftermarket vendor...at about the same price as above. I think getting the BMW branded one is preferable...just because...and it's near the same price as the non-BMW charger. GET IT!
I'd also suggest just getting a direct to battery setup. It's cheap and it's handy for charging and for plugging in a small 12V compressor to fill tires. I've used it for charging cell phones too.
Under the little triangle plastic cover on the right side of the bike (one screw to remove it on the GS), you'll find the battery in sideways. The ground can be reached by sliding the battery out a little. The positive side has a convenient post to the right of the battery as you stand looking at the under-the-triangle area (with bright red round plastic cover).
Hook the + side of your pigtail to that instead of the positive battery lead. Much tidier and I think also safer.