Difference in mechanisms between formation of acetal and enamine
This does indeed work. One such reference includes US patent 3696080A in which propionaldehyde is reacted with dimethylamine to give the N,N-acetal as shown in the figure below.

The described reaction procedure involves cooling both reagents to $\pu{-78^\circ C}$ before addition, letting the temperature rise to around $\pu{-35^\circ C}$ at which point anhydrous potassium carbonate was added, letting the temperature rise again to about $\pu{0^\circ C}$ at which point the cooling apparatus was switched from dry ice/acetone to ice/water, and then stirring it overnight allowing the ice to melt in the process. The workup the inventors describe is again rather complex but they claim to isolate up to $\pu{73\%}$ of the bis-amino product.
SciFinder also gave a number of other references but one should serve sufficiently as a proof of concept.
It’s worth noting that $\pu{2.4 eq}$ of amine were added which probably assisted in driving the reaction to the product side.