In Burris' universal algebra book:
Definition 1.2. A language (or type) of algebras is a set F of function symbols such that a nonnegative integer n is assigned to each member f of F . This integer is called the arity (or rank) of f, and f is said to be an n-ary function symbol.
Are the use of "language" in universal algebra and its use in formal languages related? (I think that they are unrelated, but am not sure if the choices of the names are only coincidence.)
A language of algebras is exactly a set of nonlogic symbols in logic, which is then used to create a formal language in logic.
They're not unrelated, but they're definitely not the same thing. A language in the universal algebraic sense should be thought of as part of the alphabet for a particular formal language, the remaining alphabet of which consists of "$=$" and a collection of variable symbols (or maybe more if we want to look beyond the purely equational). For this reason I prefer "signature" to refer to the set of function symbols in universal algebra.
(Note that the same is true for, say, first-order model theory: we have a fixed collection of logical symbols - parentheses, quantifiers, Booleans, variables, and equality - and a varying set of non-logical symbols, which form the alphabet for the formal language of wffs.)