Disclaimer: This is a long-winded attempt to answer the "... Are these guises of the same phenomenon?" part of the question. I do not understand how to translate between the two.
There is a theory of $E_n$ algebras that describes the collection of all algebras that have '$n$ dimensions worth of multiplications'. One way of making this idea precise is to take the category $\text{Disk}^n$, whose objects are disjoint unions of $n$-disks, and whose morphisms are embeddings. The operation of disjoint union gives this category a symmetric monoidal structure. If we allow for homotopies of embeddings, and homotopies of such homotopies, we can think of $\text{Disk}^n$ as a symmetric monoidal $(\infty,1)$-category.
If $\mathcal S$ is another symmetric monoidal $\infty$-category, then an $E_n$ algebra $F$ valued in $\mathcal S$ is just a symmetric monoidal functor
$$F:\text{Disk}^n\longrightarrow\mathcal S\;.$$
We often conflate the functor with the object $\mathcal C=F(\mathbf{D}^n)$ in $\mathcal S$. The fact that two disks can be embedded into a single disk gives rise to a multiplication map $\mathcal C\otimes\mathcal C\to \mathcal C$. The fact that two disks can be embedded into a single disk in multiple ways means that there are multiple such multiplications on $\mathcal C$, and the fact that some embeddings are homotopic gives rise to things like braidings, etc.
Unpacking this definition for $n=1$ and $2$, and $\mathcal S=\text{Vec}$ (=vector spaces) and $2\text{Vec}$ (= semisimple linear categories), we find the following table:
| $\downarrow n$ $\hspace{2mm}$ $\mathcal S\rightarrow$ |
$\text{Vec}$ |
$2\text{Vec}$ |
| 1 |
Algebra |
Tensor Category |
| 2 |
Commutative Algebra |
Braided Tensor Category |
| 3 |
Commutative Algebra |
Symmetric Tensor Category |
The center of an $E_n$ algebra $\mathcal C$ is an $E_{n+1}$ algebra denoted $\mathcal Z(\mathcal C)$. The definition is a bit technical, so I won't go into it here, but you can find it on page 4 of [Brochier,Jordan,Safronov,Snyder]. The center of an algebra is precisely the classical notion of center. The center of a tensor category is the Drinfel'd center. The center of a braided tensor category is the Müger center.
For $E_1$-algebras, the notions of central simple algebras, Azumaya algebras, and factors are all different incarnations of the condition that $\mathcal Z(\mathcal C)=\mathbf 1$. For $E_2$-algebras, the condition that $\mathcal Z(\mathcal C)=\mathbf 1$ is sometimes referred to as being modular, or factorizable. In general $\mathcal Z(\mathcal C)=\mathbf 1$ is related to higher Morita invertibility.
You may have heard of the fact that all Drinfel'd centers are modular. This fact is a special case of the more general idea that $\mathcal Z^2$ ought to always be trivial. Here I use the word idea instead of the word fact, because I believe that extra hypotheses need to be added to the theory in order to make this work. Either extra assumptions on $\mathcal S$, or perhaps extra structure on the disks, such as framing, oriantability etc. At any rate, for the purposes of TQFTs, a physical argument for why $\mathcal Z^2=\mathbf 1$ is given as Corollary 7 in [Kong,Wen]. There they lay out a vision of $\mathcal Z$ being the coboundary map of a chain complex
$$\cdots E_{n-2}\mathop{\longrightarrow}\limits^{\mathcal Z}E_{n-1}\mathop{\longrightarrow}\limits^{\mathcal Z}E_{n}\mathop{\longrightarrow}\limits^{\mathcal Z}E_{n+1}\mathop{\longrightarrow}\limits^{\mathcal Z}\cdots$$
In this perspective, the statement that an algebra is a factor is just the statement that it is a 1-cocycle, and the condition that a braided category is modular is just that it is a 2-cocycle.
As a follow up, I can give you two examples where the resulting cohomology groups are interesting. For semisimple $E_n$ algebras in $\mathcal S=2\text{Vec}$ (over the complex numbers), the homology group
$$\frac{\text{ker}\big(E_2\mathop{\longrightarrow}\limits^{\mathcal Z}E_3\big)}{\text{im}\big(E_1\mathop{\longrightarrow}\limits^{\mathcal Z}E_2\big)}$$
is precisely the Witt group of [Davydov,Müger,Nikshych,Ostrik].
For $E_n$ algebras in $\mathcal S=\text{Vec}$, the homology group
$$\frac{\text{ker}\big(E_1\mathop{\longrightarrow}\limits^{\mathcal Z}E_2\big)}{\text{im}\big(E_0\mathop{\longrightarrow}\limits^{\mathcal Z}E_1\big)}$$
counts central simple algebras modulo matrix algebras. In other words it is precisely the Brauer group of the base field.
