Are there any rules or clarifications that say anything about limitations or mechanical implications of having multiple characters on a single mount?
The relevant rules would be with respect to carrying capacity, or optional rules of encumbrance. It's not a fine-grained system, to simulate reality closely; it's a loose system, to facilitate game play and roleplaying without undue fuss.
So if the mount has the strength to carry both persons (without being overly encumbered if you're using those rules), then it can carry both persons.
If the saddle is not adequate, this may be ruled by the DM to require extra Animal Handling rolls if anything difficult is attempted, and maybe with a higher DC for anyone not properly in a saddle.
Not really. Here's what it does say.
There are no rules specifying what happens with multiple riders. If a rider chooses to control the mount, the initiative changes but the action and movement may have already been expended for its turn. It doesn't say the mount gets another turn only that the initiative changes.
The initiative of a controlled mount changes to match yours when you mount it
If a second rider controls the mount, the initiative again changes. The only benefit to this is staggering movement along all turns of the riders which makes sense logically too. I.E. the fighter steers the giant lizard to the trap door and batters it open with his maul. The wizard then grabs the reigns and bids the lizard to leap out so he can launch a fireball at the crowd of goblins in the room above
On your turn, you can move a distance up to your speed and take one action.
From this question:
The mount does get an extra turn; however, if it has already acted as has every other creature, the round ends before it's extra turn:
When everyone involved in the combat has had a turn, the round ends. Repeat step 4 until the fighting stops.
Can the mount carry two?
You also need to be mindful of whether the mount is physically able to accommodate multiple riders. This involves carrying capacity (including accounting for variant encumbrance if the table uses it), but also revolves around the creature itself.
A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount, using the following rules.
The appropriate anatomy may only be appropriate for one character and, as such, fails to be appropriate when a second character attempts to utilize the mount.