Why do we think that there is no two-solar-mass black hole? We think that the mass boundary between neutron stars and stellar mass black holes is  around three solar masses. 

The maximum mass of the neutron stars now is two solar masses and we may find a 2.6 solar mass neutron star in the future. That is based on the equation of state.

The stellar mass black holes may have different origins. Even the smallest stellar mass black hole originates in SN explosion,  we do not know the details of the explosion.

Why do not we think a two solar mass BH, or even a smaller BH, could exist?

Why do we think that the companion of Taylor-Hulse binary is a neutron star instead of a BH?
As you said in your question we only know about Black Holes creating in Super Novae, which lead to relatively big BH. The reason they can't be smaller in these cases it that they are created when the core of the star reaches a critical mass that overflows both the electron and the neutron degeneracy. Lower masses can be hold by those degenerate mass states and won't collapse any further.

Theoretically BH with masses less than the solar one could exists if there was some method capable to compress it's mass reaching radii smaller than their Schwarzschild radius.