Here's my understanding of things.

Chocobos take up space in a party, but they are not necessarily considered "Party Members". They're almost like Guest NPCs, in a sense. If you form three players and a chocobo together, you don't get access to Limit Breaks whatsoever, because a Chocobo does not count as a party member, and thus, the party is not considered a "Light Party", which is required for four-man dungeons. This also has complications in the duty finder because, if a party is not a "Light Party", then it would allow players to recruit. However, if there are already four in the party (three and a Chocobo), then you can't have any more characters in the party in the first place.

Chocobo companions, at least when it comes to party makeup, is kinda paradoxical like that. And I'm sure, the way the system was developed, it would require some serious retweaking of the companion system for it to work to accommodate Chocobos (and maybe Magitek Armor later on) in parties for duty finder.

I know it's not impossible because SWTOR did it, but the difference there is that SWTOR built their party system with companions in mind. FFXIV did not. So... yeah. Definitely gonna need some serious code tweaking to fix companions in parties.