It seems numerically that this equality holds for $n_1,n_2 \in \mathbb{N}$ :
$\frac{\Gamma (n_1+1) \Gamma (n_2+1)}{\Gamma (n_1+n_2+2)} = \frac{1}{2^{n_1+n_2+2}} \left(\Gamma (n_1+1) \, _2F_1^{(reg)}\left(1,n_1+n_2+2;n_1+2;\frac{1}{2}\right)+\Gamma (n_2+1) \, _2F_1^{(reg)}\left(1,n_1+n_2+2;n_2+2;\frac{1}{2}\right)\right)$
But I don't manage to show it using the properties of the regularized hypergeometric function.
Any help would be welcomed, thanks !