Using the books' rules, no, you can't.

You can only choose your archetype once, when you achieve that level (e.g. 3rd lvl Fighter). The only way to get another archetype would be leveling the same class from 1st level again, doing something similar to a multiclass Fighter/Fighter.

Although, as mentioned here, multiclassing rules in PHB specifically state

With this rule, you have the option of gaining a level in a new class whenever you advance in level, instead of gaining a level in your current class.

And in Sage Advice

Can I multiclass into the same class to pick more than one subclass?

Multiclassing is designed solely for taking different classes. For example, you can be a fighter/rogue, but not a rogue/rogue.

(Just explaining to people that might get here to question the same thing and don't know why it is against standard rules, as mentioned in the text)

Hopefully, the books aren't everything

I know the same about specific official guidelines to multiple archetypes as you - I don't know any. And yet, I've created some NPCs and let my players create some PCs with multiple archetypes, although admittedly some turned out being broken. Anyway I might help:

• Create a mixed archetype that takes out some features from one archetype and adds from the other.
• Your scenario seems motivated by the background and race of the character, so it seems you want the most flavorish characteristics. From Wild Magic, you want Wild Surge and from Draconic Bloodline you want the Draconic Resilience.
• Balancing both of them together: If you give them the Tides of Chaos from Wild Magic as well, decrease the AC given by draconic resilience and don't give them the HP bonus. Also nerf some of the wild surge effects from the table as you seem fit.
• If you give them the full Draconic Resilience, don't give them Tides of Chaos. Honestly, I think Wild Surge by itself is not a "good" feature that actually needs to be balanced if you just give it for free. For example, casting a Fireball centered on yourself usually means killing yourself and half your party. There are enough negative effects already and its trigger is the DM wants to, so you can actually just give it for free and balance it through when you trigger it.

Personally, I would make them take the Draconic Bloodline archetype and just give the Wild Surge feature for free. Balance-wise it is does not seem broken to me, although I do have less experience with Wild Magic Sorcerers than almost every other subclass.

You’re missing the two-weapon fighting combat option rules. Without those, the feat et al. won’t make very much sense.

The two-weapon fighting combat option allows you to get an extra attack when you full attack, but all of your attacks in that full attack take a penalty. The extra attack has to be with a different weapon from your regular attack (“offhand” weapon), and the penalties you take depend on whether or not that offhand weapon is light or not.

The Two-Weapon Fighting feat reduces those penalties to a point that this option is actually usable.

Another key thing to realize is that having two weapons in hand does not automatically mean you are using the two-weapon fighting option. You have to choose to use it, to get an extra attack and take an attack penalty.

As for the +6/+1 thing, that’s called an iterative attack, and that’s free just for having BAB that high. It is separate from, and can be combined with, the two-weapon fighting option. So, assuming you have the Two-Weapon Fighting feat and a light weapon in the offhand (the only case in which the two-weapon fighting option is a good idea), with +6/+1 BAB you could attack normally, for +6/+1, or you could choose to use two-weapon fighting, taking a −2 penalty. Then your attacks would be +4/+4/−1, where the second +4 attack has to be with the offhand weapon. If you take the Improved Two-Weapon Fighting feat, that becomes +4/+4/−1/−1, because Improved Two-Weapon Fighting gives you a second bonus attack with the offhand weapon, at −5 (matching the penalty on the iterative attack).