Dedicating two identical mini-pets in different characters, under the same account

They are indeed mutually exclusive; everything you stick in your hall will only count once, including mini pets. You'll need 50 distinct ones to get all the points for it, as well as distinct armor sets for the armor monument and all heroes and special pets displayed in that particular monument as well.

I'm going to go on the assumption you want the access to the Hall of Monuments for your Guild Wars 2 playthrough. In this case, your best bet would be to start a PvE character in Factions or Nightfall, if you want to do it through normal play. However, there's a few things you may want to consider, depending on what you plan to do.

For reference's sake, you can get there by talking to the following characters. You must be at least level 10 in order to accept any of these quests:

• Len Caldoron in Lion's Arch (Prophecies)
• Minister of Maintenance Raiugyon in Kaineng City (Factions)
• Bendah in Kamadan (Nightfall)

Considering how slow Prophecies plays in regards to leveling curve and mission pacing, this isn't an ideal choice at all really. You could pay to have someone run you to Lion's Arch, but you may not be level 10 by the time you get there. As I've never done the run myself, I'm unsure. However, Prophecies has the easiest obtainable weapons for the HoM because of Oppressor weapons, the requirements for which are less work and cheaper than Tormented and Destroyer weapons.

Factions' leveling curve to 20 is the fastest. Even without taking the sidequests on Shing Jea Island into consideration, you should be a good chunk of the way to level 20 by the time you finish the missions there and travel to Kaineng City. Factions also has the shortest campaign, so you could finish that out for some easy points for your HoM, in addition to 2 of the unique pets you need for the Monument of Fellowship, the Black Moa and the Phoenix.

However, Nightfall lets you receive a bunch of heroes (Koss, Dunkoro, Melonni, Tahlkora, and Acolyte Jin/Acolyte Sousuke) pretty early, which will make completing more of the PvE content easier, if that's your plan. In addition, the Nightfall campaign itself has the most points to offer because of Vabbian armor, Tormented weapons, and the most heroes.

Based on these factors and depending on your long-term goals, I'd say it's a toss-up between Factions and Nightfall. If you just want to just level up to get straight to EotN and don't care about the Nightfall heroes, go Factions (as you do still get Vekk, Gwen and Ogden in EotN), but if you want to have more heroes for PvE play, start in Nightfall.