I know that an algebra is an algebraic structure, that can be seen as a vector space with a multiplication operation or as a ring with a vector space structure. However, in measure theory we define an algebra as a collection of subsets closed under finite unions and intersections.
My question is: Are these "algebras" related or do they have anything else in common except their name?
Yes, this is the same notion. Note that an algebra is also closed under complement and contains the whole set and the empty set.
If ${\cal A}$ is an algebra of subsets of some set $X$, we can define on it two operations: the symmetric difference $\Delta$ and the intersection $\cap$. Note the relations between the following indicator functions: $$ {\bf 1}_{A \Delta B}(x) = {\bf 1}_A(x) + {\bf 1}_B(x) \ \ \hbox { mod } 2, $$ $$ {\bf 1}_{A \cap B}(x) = {\bf 1}_A(x) {\bf 1}_B(x). $$ So we have an explicit injective morphism between the algebra $\cal A$ endowed with the two operations $\Delta$ and $\cap$ and the algebra of functions from $X$ to ${\bf Z}/ 2{\bf Z}$ endowed with $+$ and $\times$. It is given by $$ \matrix{{\cal A} & \mapsto & {({\bf Z}/ 2{\bf Z})}^{X} \cr A & \mapsto & {\bf 1}_A \cr} $$ As a result, $\cal A$ is isomorphic to the image of this morphism as an algebra.
