There are some multi-seat mounts available from in-game content... but they're difficult to get - there's the Blackjack airship from FFVI on the Gold Saucer which is a four-seater mount, but that costs a whopping 3 million MGP (took me over a year to save up enough to get, although I was playing very casually), and if you do the 'Recruit a Friend' campaign, you can get the Amber Draught Chocobo which is a dual-rider mount. There is also the Manaslipper mount from Eden Savage which is a four-person mount, but the drop rate is... low.

Purchasing the Gold or Platinum Ceremony of Eternal Bonding also gives you the dual-rider Ceremony Chocobo, although that is technically a paid service (if your partner pays for it you get the chocobo for free).