Classes and Algebras: Which begat which?
I'm not sure I fully understand your question, but I think you noticed that algebras (or more precisely algebraic signatures) and classes share a feature, namely that they both can be seen as specifying a bunch of operations. For example the algebraic construct group has three operations: multiplication, inverse and the constant $1$. Now you ask if algebras can be used to model classes.

The answer is negative at this point. Classes in the sense of OO programming are complicated beasts. They have features that algebras don't have, in particular computational dynamics. There are other features like inheritance, method overriding, mutable private members that also
don't lend themselves to easy algebraic description. For this reason, purely algebraic sematics of OO languages don't exist. 

As to the historic question, I don't know. The development of types goes back to the end of the 19th century, a time when abstract algebra was in nascent form too