Since the Pseudodragon is still CR 1/4 I don't see there being too many problems with this. I don't think the creature's type is going to make any huge difference regarding balance or gameplay.
The main things that jump out at me from its stat block are that it can do extra poison damage on a save, but then so can the Giant Centipede, which is a CR 1/4 beast (and probably other CR 1/4 beasts, but that's the first one I found), so that's fine. The other is the resistance to magic (meaning that it has advantage on saving throws).
In the Monster Manual pg 254, there is mention of a Variant: Pseudodragon Famaliar, which allows the PC to share the Magic Resistance trait. I would not allow this as this particular Pseudodragon isn't strictly a familiar, it's a "beast" companion via the Beastmaster class feature (I assume that's how your player is going to be getting a beast companion?).
So avoiding the Variant should keep this balanced (since having free advantage against magic would most definitely make this an overpowered alternative to standard CR 1/4 beasts).
Regarding what happens if it were to die, the PHB (pg 93) says for the Ranger's Companion:
If the beast dies, you can obtain another one by spending 8 hours magically bonding with another beast that isn't hostile to you
So if you're happy for another Pseudodragon to turn up when they're about to take a long rest, then become friends with the Dragonborn ranger during that long rest, that would seem balanced enough (although technically it wouldn't be able to help with any fights until the long rest had been completed, since they haven't "bonded" yet).