A string representation of any Turing machine
Consider a Turing machine with $n$ tape symbols and $m$ states. We can assume that the symbols and the states are ordered, so that we can talk about the first, second, ..., $n$th tape symbol, and the first, second, ..., $m$th state. (This is similar to your idea of counting Turing machines up to isomorphism.) So without loss of generality, the tape alphabet is $\{1,\ldots,n\}$ and the states are $\{1,\ldots,m\}$. We will encode this Turing machine as the following sequence of statements in English: (the notation $\langle n \rangle$ means to replace $n$ with its decimal representation)


The number of tape symbols is $\langle n \rangle$.
The number of states is $\langle m \rangle$.
The starting state is $\langle s_0 \rangle$. [Replace $s_0$ with the starting state]
The accepting states are $\langle s_1 \rangle$, ..., $\langle s_k \rangle$. [Replace $s_1,\ldots,s_k$ with the accepting states]
When in state 1, upon reading symbol 1, write symbol $\langle \sigma \rangle$ and move $\langle D \rangle$ [Replace $\sigma$ with the tape symbol and $D$ with either "left" or "right"]
...


This is a description in English over a finite alphabet. In particular, it can be represented in ASCII. So there are only countably many descriptions.



As usul mentions, we tacitly assume that the underlying representation of everything in mathematics is a set, just like the description of everything in computers is in binary. Not all sets are finite, but if you only use hereditarily finite sets, then you can avoid forcing the tape alphabet to be $\{1,\ldots,n\}$, and instead allow arbitrary hereditarily finite sets. However, that doesn't buy you anything (unless you care about this point for philosophical reasons).

More generally, you might be worried that when I describe a Turing machine I use the symbol $\diamond$, a symbol not covered by the above (it's a set-theoretic "atom"), so this Turing machine doesn't have a description in my formalism. To counter this, I suggest you provide the description of your Turing machine as a LaTeX document (which is what I used when describing the machine in the first place). LaTeX has a finite alphabet (or finitely many possible finite alphabets), so again everything is countable.