Nothing happens; you can't double Meddling Mage's 'enter the battlefield' trigger since it doesn't have one.
But then... what is the thing that happens as the Mage enters the battlefield? Well, it's a replacement effect:
614.1c Effects that read “[This permanent] enters the battlefield with . . . ,” “As [this permanent] enters the battlefield . . . ,” or “[This permanent] enters the battlefield as . . . “ are replacement effects.
while triggered abilities are always worded differently (at least in Oracle, but Meddling Mage is modern enough):
603.1. Triggered abilities have a trigger condition and an effect. They are written as “[When/Whenever/At] [trigger condition or event], [effect]. [Instructions (if any).]”
The rulings below Panharmonicon also note this:
Abilities that apply “as [this creature] enters the battlefield,” such as choosing a creature to copy with Clone, are unaffected.
In most Magic games, the difference between these kind of replacement effects and triggered abilities can be neglected, but not in this particular case. Another difference is that after Meddling Mage starts to resolve and it's on the battlefield 'banning' a specific card, nobody gets priority so you can't sneak in a final casting of the named card - you'd have to do so while the Mage is still on the stack.
Glorfindel's answer correctly covers the case of Meddling Mage and Panharmonicon, but to answer what happens if you did manage to choose two different card names for a similar ability:
This could either happen if the first ability actually were a triggered ability instead of a replacement effect; or if there were some replacement effect (which does not exist on any actual Magic cards) such as "if you would choose a card name, choose two card names instead".
If you did choose 2 different card names with such an ability, the second ability would refer to either of the card names; and your opponent would be unable to cast cards with either name.
The 2 abilities printed on Meddling Mage are called linked abilities:
- Linked Abilities
607.1. An object may have two abilities printed on it such that one of them causes actions to be taken or objects or players to be affected and the other one directly refers to those actions, objects, or players. If so, these two abilities are linked: the second refers only to actions that were taken or objects or players that were affected by the first, and not by any other ability.
And more specifically:
607.2d If an object has an ability printed on it that causes a player to “choose a [value]” and an ability printed on it that refers to “the chosen [value],” “the last chosen [value],” or similar, those abilities are linked. The second ability refers only to a choice made as a result of the first ability.
In the case of Meddling Mage; the second ability is referring to any card name that was chosen by the first ability.
Note that certain cards such as Shapeshifter use the phrase "the last chosen" instead of just "the chosen"; to differentiate abilities where you would not be able to get the second ability to apply to multiple choices made by the first ability.