Are real numbers defined as Banach spaces? I was listening to this lecture by Norman Wildberger, where he explains that real numbers are constructed as equivalence classes of Cauchy sequences. He never says that these equivalence classes are Banach spaces, but I was left wondering if that is what he means.

This is unfortunately a nonsense question.
There is a set of Cauchy sequences. This set has a relation on it ($(a_n) \sim (b_n)$ if $\lim_n (a_n - b_n) = 0$). There is therefore a quotient set or set of equivalence classes.
None of this ever passes through anything with more structure than sets. (If you like, it passes through rings: the set of Cauchy sequences is a ring, because I know how to add and divide in it, and the relation respects these operations, so that the quotient inherits them.)
In fact "$V$ is a Banach space" does not even make sense until you invent the real numbers. A Banach space is a pair $(V, N)$, where $N: V \to \Bbb R_{\geq 0}$ is a function satisfying certain properties.
