see the attachment for a picture of the board. Looks like a CR1220 battery to me:
https://www.ebay.com/itm/20-Pcs-2-Pins-tabs-CR1220-1220-3V-Cell-Coin-Battery-Right/222869994359your picture is a prototype, they probably used different components for that

You can often find these shrink wrapped batteries in Laptops for the CMOS. Technically you could probably also shrink wrap the battery yourself using shrink tubing, but that seems only a reasonable option if you're soldering often and already have the materials in stock. Not something for a "one-off-job".
EDIT: Either way: Yes, the battery is soldered in place on the omega, and yes, it's definatelly a lot more comfortable on the EDGBA with its battery holder. And while it isn't too hard for someone with rather basic soldering skills to replace the battery, it's not for everyone and it's understandable if someone who doesn't have the time or patience (or doesn't feel comfortable soldering) or simply doesn't have a soldering iron doesn't want to deal with that.
EDIT 2: I also just realized that the image of the Golden Sun cartridge you posted is actually a counterfeit card. Original GBA games don't have that black blob on them. Also, Golden Sun uses Flash storage for its savegame and therefore wouldn't require a battery. Just a bit of information I noticed
