When reading mathematical writing from the time of Frege and Russell, I frequently see the use of the word 'class', and as far as I can tell they mean it in the sense modern mathematics use the word 'set'.
When I try to look up the meaning of the word 'class', I can only find it in terms of modern theories like ZFC.
Is it correct to interpret the word 'class' as used by Frege and Russell as meaning the same thing as 'set'?
Pick up a copy of Russell's "Principles of Mathematics" to know what Russell said. You will find that he distinguishes between classes, class-concepts, and collections. Collections are associated with mathematical contexts. They are, in effect, enumerations of individuals. By contrast, classes are aggregates of individuals. In so far as the modern notion of set is sometimes said to be a class that is an element of another class, Russell's classes ought not be compared with modern sets. Class-concepts require an understanding of the difference between intensional logic and extensional logic as it had been inherited from Leibniz.
When Frege declared that logicians ought concern themselves with truth, he effectively introduced semantics as opposed to epistemology. By invoking the "function concept" of mathematics, he introduced a distinction between an intensional name for a concept and the extension of a concept. Individuals are said to "fall under" concepts. This is where naive comprehension arises in the logicism of Frege.
With regard to. modern paraphrasing, Frege simply recognized the importance of changing a statement that uses a proper name into a formula where the proper name is replaced by a variable. Eventually this leads to the specification of symbols into syntactic categories when describing some particular formal system. One must at least distinguish between proper names and variables.
The influence of Leibniz lies with his portrayal of logic using an inversion of the order in the Aristotelian class hierarchy. Aristotle had considered grammatical subjects. The names of individuals correspond with "primary" substances. Other subjects, now associated with the Russellian class-concept or some modern accounts of types, correspond with "secondary" substances. Our understanding of the subset ordering corresponds with this "directionality."
Leibniz recognized that logic could be inverted because orders have converses. His influence for this appears to have been St. Thomas Aquinas. St. Thomas argued that God could know individual souls because individuals are the smallest (extensional) species. This is the source of Leibniz' identity of indiscernibles. Today we speak of filters and ideals.
Frege saw classes and extensions of classes.