Do amphoteric cations exist?
Yes, for example lysine.

Lysine has two amino groups and one carboxyl group. When dissolved in aprotic solvents, all of these will be uncharged so the carboxyl group will be $\ce{-COOH}$ and the two amino groups $\ce{-NH2}$. When dissolved in water, these three groups have distinct $\mathrm pK_\mathrm a$ values of $2.15, 9.16$ and $10.67$. Depending on the acidity of the solution, the following (acid-sided) equilibriums occur:



When dissolved in water, the carboxyl group’s proton will immediately be removed and used to immediately protonate the &epsilon;-amino group (the far end). However, the &alpha;-amino group is still more basic than water and thus will be partially protonated; meaning that the right-hand equilibrium will be happening predominantly if lysine is dissolved in water. Dissolved lysine will be a mixture of uncharged lysine and lysine cation.

When an acid such as hydrochloric acid is added, at first the &alpha;-amino group is completely protonated and then the left-hand equilibrium occurs predominantly as more and more lysine cation is converted to lysine dication.

If a base is added (such as sodium hydroxide), the &epsilon;-amino group’s proton can also be removed yielding lysine anion; I have not shown that equilibrium in the scheme above.

Lysine cation thus fits your question as it is an amphoteric cation, the conjugate acid of a dibasic compound.