Can I rewrite the convex optimization problem
\begin{align} &\min \,\, f(x_1,x_2) \\ &\text{s.t.} \, \, x_1^2 + x_2^2 \leq r \end{align}
with $r \in \mathbb{R}$ as a conic optimization problem? As far as I know, the set described by the constraint is not a cone, since for example it is not closed w.r.t. addition or a scalar multiplication. However, I found the post where a problem with a similar quadratic equality constraint is described as conic. What is the difference?
For a problem to be conic, you do not require $x$ itself to be in a cone, but rather $Ax+b\in K$, where $K$ is a cone. The constraint in your case is
$$\begin{pmatrix}1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}x + \begin{pmatrix}0 \\ 0 \\ \sqrt{r}\end{pmatrix} \in K$$ where $K$ is the second order cone.
Whether the problem is conic depends on the objective function.