@IrisTherian; in general companions can only be obtained by buying them.
1. Mole Market: the vast majority of companions can (only) be found here.
2. Arena market: Some companions can be bought from the arena shop with arena tickets. However, only a couple of them are actually exclusive to this shop. Most of them can still be found in the mole market.
3. Weekly event shops: each weekly event shop stocks 4 companions exclusive to that shop
4. Achievements: Once you've levelled up your companions a bunch, there are a few companions that are awarded through the achievement system at certain milestones.
5. Referrals: For some reason the staff also introduced non-tradeable, 1-per-account companions 100% exclusive to the referral system. Several people need to sign up using your link before you can claim all five of these.
6. Pack events: Four times per year a month-long event will take place that will contain several companions. Some are exclusive to that event (e.g. returning every year), some are exclusive to that year's event (e.g. never returning to purchse outright), and some can be bought during each of the four events. October 1st the next pack event will start. Again all these companions will need to be bought (with event tickets)
Sorry for any extra pings. I made a few edits cause I made some weird typos.