Solution:

Let $\underline{a} \ \underline{b}$ be a $2$-digit cuddly number.

Then $$10a + b = a + b^2.$$ Rearranging, we get $$9a = b(b - 1).$$ This means that $9$ divides either $b$ or $b - 1$ ($3$ cannot divide both $b$ and $b - 1$).

The only way this is possible is if $b = 9$ ($b$ is one digit, so it can't be anything else). Checking, we get that $89$ is a cuddly number. This shows that there is only $1$ two-digit cuddly number.

Thus, B is the correct answer.