Although the Battle Pet Trainer will only offer to teach you your race-specific companion, it is possible to add all of these pets to your collection. You can either find them in your faction’s auction house or you can create a character of the race whose pet you would like to learn. Using your new character, you can seek out any battle pet trainer in major cities. After you learn the pet, you can delete the character; the pet you learned will not be removed from your collection.
Other methods have been discussed. For example, mods could be crippled. DBM didn't exist back then, and the other boss that did that I don't remember the name of, didn't do as much back then as DBM does now. If our goal is to recreate the experience as it was, then obviously not having DBM would be a reasonable way to accomplish that. DBM simply existing, makes the game today easier than it was. So disallow it. Again, this is completely consistent with the spirit of vanilla. Remember one-button-decursing? That was nerfed. They crippled the capabilities of the modding API to disallow that because it made the game too easy. So what if they cripple it a little bit more to once again make the game less easy?
FYI: If you’re returning to WoW after an extended absence prior to patch 5.0.1, it’s possible that you’ll have more than the maximum pet limit and/or more than 3 of a kind. However, this will be the only time that you’re allowed to surpass these limits. You will have to pare down your collection, eliminating your duplicates, in order to add new pets.