AucTeX redefines the binding for `$` to be the command `TeX-insert-dollar`. That command checks whether you are in a math environment already and complains if you try to close it the "wrong" way. Doing `C-h f TeX-insert-dollar` shows you the doc string of the function:

```
Insert dollar sign.

If current math mode was not entered with a dollar, refuse to
insert one.  Show matching dollar sign if this dollar sign ends
the TeX math mode and ‘blink-matching-paren’ is non-nil.

When outside math mode, the behavior is controlled by the variable
‘TeX-electric-math’.

With raw C-u prefix, insert exactly one dollar
sign.  With optional ARG, insert that many dollar signs.
```

Note the part that says: **If current math mode was not entered with a dollar, refuse to insert one.**

BTW, it's not clear to me what the `\[ ... \]` delimiters are buying you. If you get rid of them, then AucTeX will not complain about the `$` signs, and  you still have a centered display. CORRECTION: It's not centered but you can do that with `\begin{center}...\end{center}`.

Alternatively, keep the display but change the font of the text of the nodes:

```lang-latex
\[
  \begin{tikzpicture}[every text node part/.style={font=\itshape}]
    \node (X) {X};
    \node[right of= X] (Y) {Y};
    \draw[->] (X.east) -- (Y.west);
  \end{tikzpicture}
\]
```