Closed Form Solution of $ \arg \min_{x} {\left\| x - y \right\|}_{2}^{2} + \lambda {\left\|x \right\|}_{2} $ - Tikhonov Regularized Least Squares

That's not what the referenced paper says. It gives an expression which is equivalent to the proximal operator of the $\ell_2$ norm:

$$ \DeclareMathOperator*{\argmin}{arg\,min} \argmin_x \frac{1}{2}\|x-y\|^2 + \lambda\|x\| = \max(\|y\|-\lambda,0)\frac{y}{\|y\|} $$ Note the vector $y$ is not inside the maximum.

I'll sketch a proof. We can decompose $x$ as sum of two components, one parallel to $y$ and one orthogonal to $y$. That is, let $ x = t \frac{y}{ \| y\| } + z $ where $y^T z=0$. Then the objective reduces to:

$$\frac{1}{2}\|x-y\|^2 + \lambda\|x\| = \frac{1}{2}\|z\|^2 + \frac{1}{2}(t-\|y\|)^2 + \lambda \sqrt{t^2 + \|z\|^2}$$ Clearly the expression is minimized when $z=0$, so the problem reduces to a 1-dimensional problem: $$ \min_t \frac{1}{2}(t-\|y\|)^2 + \lambda |t| $$ Then it's a basic exercise in calculus to show that the objective is minimized when $t=\max(\|y\|-\lambda,0)$.