Only if they are equivalent to the new party.
In general you should only allow a player to bring in an existing character if they are on par with the other members of the party.
As a DM that uses the milestone leveling system I dislike my players being at different levels. I wouldn't suggest allowing him to join a level one party at level two as it will permanently imbalance your party. Either level up the others to match or make him return to level one.
This is the bit that sounds fishy. I wouldn't allow a player to join with any bonuses from a different campaign. Recalculate his max hit points based on class and constitution and tell him that in this world his potions didn't exist.
The +7 to Initiative sounds like they have taken the Alert Feat from the Player's Handbook. The feat states:
- You gain a +5 bonus to initiative
- You can't be surprised while you are conscious
- Other creatures don't gain advantage on attack rolls against you as a result of being hidden from you.
The only way they could have this feat at level 2 would be by being a variant human however so check that.
Players who like to play the same character often encounter this situation. It's possible he became attached to this character and didn't get to continue his story, either make his previous story canon or not. Up to you and what will suit your world. You can rule that this is an alternate reality and while he is the same character he doesn't have any of the experiences he had previously.
Make it equal
In general, you can allow players to play anything you like so long as the party is even and fair. Make sure he is playing a character created within the rules and is even with the other players.