Gaining resistance at level 1 is ok, but gaining immunity is (too) strong.
There aren't many effects that grant immunity to a damage type. A few spells do something like that (Investiture of Ice/Flame, Mind Blank, and of course the Invulnerability spell), and level 10 monks or land druids get poison immunity, as do Yuan-Ti Purebloods from XGtE. I'm sure there are a few others that I missed, but they're probably similar to those above in terms of cost or impact. See also Barring Epic Boons, is there a way to gain immunity to fire damage?, which (among others) lists a few options for becoming immune to fire - though all very high level or requiring legendary magic items.
That being said: these options are all either expensive or only very specific and available to a very small subset of players.
Now, I haven't personally played with your house rule, and thus I can't judge from personal experience. Based on my general expertise with 5e and my subsequent bonus to WIS rolls, however, I would say that it's either overpowered, or as strong as you can get without being overpowered.
With your houserule, the player has more flexibility in picking the damage type than with other abilities, although (s)he can only do so at the start and is then locked in for the remainder of the campaign, so it's not too flexible. The mechanically speaking best choice is probably fire, so that one can ignore common damaging spells like Burning Hands, Fireball, or others (fire is one of, if not the most common damage type). As fire is the most versatile and strong choice, I will focus on it from now on.
However, if we compare your ability to other abilities that grant fire damage immunity (see above), it's evident that they are usually much higher level than level 6. This leads me to believe that gaining immunity to fire at level 6 is indeed overpowered, as it allows you to ignore the most common elemental damage type, which is useful both against enemies and for ignoring your party colleagues' fireballs, allowing for easier positioning.
Good news: it's fixable.
First off, I would change the ability to only grant immunity if your Draconic ancestry matches your Dragonborn player race. This seems to be your intent anyway, so let's just phrase it like that:
Starting at 6th level, when you cast a spell that deals damage of the type associated with your draconic ancestry, [...]. At the same time, you can spend 1 sorcery point to gain resistance to that damage type for 1 hour. If you are a Dragonborn with the same Draconic Ancestry as your Draconic Bloodline, you instead gain immunity to that damage type.
Next, gaining immunity is clearly usually reserved for higher levels, so let's modify it some more:
If you are a Dragonborn with the same Draconic Ancestry as your Draconic Bloodline, you can instead gain immunity to that damage type, starting at level 12.
I chose level 12, because while it's stronger than for example a druid's primordial ward, it's not quite as strong as some other classes' fire immunity, considering it's not permanent. Also, like level 6, level 12 is one level after advancing to a new tier of play.
If you decide to instead up the cost of gaining immunity to (significantly) more than one sorcery point, you could get away with leaving it at level 6.
Furthermore, since you are the DM, this ability will always only be as strong as you allow it to be. If you decide not to throw any enemies with fire damage in their repertoire at your party, then you probably don't have to balance the ability at all beyond your original phrasing. In that case, it will only be useful for granting the other party members more flexibility with fireball positioning and the like. However, avoiding all enemies with fire damage is next to impossible, as it eliminates very many enemies (including basically all wizards).
For balancing, you could also put more focus on the effects of being subjugated to fire while immune. Normally, the risk of getting your clothes and other possessions burned off when hit by a fireball is ignored, because it would be very annoying for the players. In this case, however, you could emphasize that effect for the Sorcerer to increase the cost of getting hit by fire (even if only marginally). In fact, it's probably a fun addition to have that one guy in the party who's always naked, yet physically fine, after getting hit by fire. It goes without saying that your player (and players - not everyone wants nudity in their game, even if it's not sexual) should be ok with this before starting the campaign.