Of course not.

But after reading a bit, some points make me believe it should be:

Let $S$ be a nice$^{\*}$ surface defined over $Spec\ \mathbb{Z}$.

- The Brauer group $Br(S\otimes \bar{\mathbb{Q}})$ is an abelian divisible group,
- It is also a $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$ module,
- For good primes there are reductions $Br(S)\rightarrow Br(S\otimes \mathbb{F}_q)$,
- These $Br(S\otimes \mathbb{F}_q)$ are finite,
- There is a formal Brauer group $\hat{Br}(S)$ of dimension 1,
- The coefficients of $\hat{Br}$, in suitable natural coordinates, relate to $|Br(S\otimes \mathbb{F}_q)|$.
- There are some examples where the associated L-function comes from a modular form (of weight 3). I'm not sure if this is conjectured (let alone known) in general.

Since the Brauer group observes many characteristics of an abelian variety (all properties) of dimension 1 (properties 5 and 7 [weight isn't two, but it's the right space]), my vague question is: how far is it from actually being a variety?

There are some easy examples of $S$ with $|Br(S\otimes \mathbb{F}_q)|$ varying between $1$ and $4(q-4)$, as $q$ varies over the primes. This is a clear point of departure from elliptic curves and varieties in general.

Maybe there's a family of natural galois-module homomorphisms into certain abelian varieties defined over $\mathbb{Q}$, commuting with the reduction maps and restriction (or some other appropriate term) to formal groups?

What's going on with these Brauer groups?

$^\*$ say a K3 surface. Something that (1) is true for (so not a rational surface) and (4) is proven for.