So, this came up in a few of the campaigns I have been part of, and I am unsure of how this should be resolved. Within the rules of the game would say a Silver Dragonborn Sorcerer with the Dragon Bloodline Origin be confined to the Silver Bloodline, or would they be allowed to have Gold power for their Sorcerer Powers?
You can indeed choose one Dragon Ancestor for your sorcerer's class feature and a different one for your dragonborn's racial trait.
A dragonborn sorcerer can choose a different ancestor for the racial trait and for the Dragon Ancestor feature.
Note that the dragonborn's description states that the species had descended (at some point) from dragons, whereas the draconic bloodline sorcerer's description is as follows:
Your innate magic comes from draconic magic that was mingled with your blood or that of your ancestors. Most often, sorcerers with this origin trace their descent back to a mighty sorcerer of ancient times who made a bargain with a dragon or who might even have claimed a dragon parent. Some of these bloodlines are well established in the world, but most are obscure. Any given sorcerer could be the first of a new bloodline, as a result of a pact or some other exceptional circumstance.
As such, the draconic bloodline sorcerer may not necessarily have a "dragon parent", but were touched by dragon magic in some other capacity. Really, you can flavor it however you want.
There is no rules reason you'd need to pick the same dragon type for both.
Certainly it would be thematic to do so, but from a purely mechanical standpoint, picking two different dragon types gives you more resistances (one all the time, from your race, and one that you can turn on with a sorcery point). And since your breath weapon wouldn't benefit from Elemental Affinity (as it's not a spell), there's no real benefit to overlapping.