Can "dirty" brick finish be removed?
what is this kind of finish/texture called in the trade?


It depends on the part of the world you live in and the manufacturer.

There are thousands of variations available, bricks tend to be classified primarily by colour but also by texture and by a lot of other attributes such as "dragface", "handmade", "thrown", "waterstruck", "extruded", "light texture", "heavy texture", "smooth" and so on ad-infinitum (only limited by the imagination of marketing folk).


  Can it be cleaned off or thinned out to reveal more or all of the base brick color underneath?


No. It is more appropriate to buy bricks in the style you want.


  Would pressure washing or sand blasting be possible or would that run the risk of harming the mortar?


Either of those would damage the bricks or mortar.