Answer: B

Solution:

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

Then $$10a + b = a + b^2.$$ Rearranging, we get $$9a = b(b - 1).$$ Because consecutive integers cannot both be divisible by $3,$ one of $b$ and $b-1$ must contain both factors of $3.$

For a digit $b,$ this leaves $b=0,1,$ or $9.$ The first two choices give $a=0,$ which does not make a two-digit number. If $b=9,$ then $a=8.$ 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.