It says that you get 1 BGA token per month when you "subscribe", but subscribing means what exactly? Buying and using a GA medal or purchasing the "value pack" subscriptions?

The latter, signing up for a monthly subscription. As the page shows, you must subscribe to the Councilour or Goddess plan to get the BGA(s).

Wow, it costs $10 to get a single token and there's no way to get them for free.

Whoever gave me the BGA had to be incredibly generous.

You can either subscribe and be a gold member for 3$ a month. You can buy a gold account medal with CSC for 500 or in SP depending on if they are listed to be sold in user shops. The buddy gold accounts are a bonus for people to give away since the subscriber cannot use it on themselves. Lots of subscribers give them away to people they randomly see blue or to specific friends they know cannot get one otherwise. Some people also just stock up and then give them away during quest-a-thons or Luminaire.