What is this set notation called?

In general, if $A$ and $B$ are subsets of some set with an addition structure, you can write $A+B = \{a+b \mid a \in A, b \in B\}$. Formally what the professor wrote is $$\beta + 2\pi \Bbb Z = \{ \beta + 2\pi n \mid n \in \Bbb Z \},$$and what you're proposing is also correct: $$e\Bbb Z + \pi \Bbb Z = \{ae+b\pi \mid a,b \in \Bbb Z \}.$$The point is precisely that $e\Bbb Z + \pi \Bbb Z \neq (e+\pi)\Bbb Z$. I do not think that this notation has any specific name, though.

Actually, you are correct: $\theta=ae+b\pi$ for any $a,b\in \mathbf{Z}$ can be written as $\theta\in \mathbf{Z}e+\mathbf{Z}\pi$. This notation simply means that if we choose any integers $n,m$ from $\mathbf{Z}$ in the first and second space, respectively, we have that $\theta=ne+m\pi$. Note that this does not require $n=m$. I'm not sure of a specific name for this notation, but it is often used in algebra.