For comedy there is a genre, for horror there is a genre, for action there is a genre.
Then why is there no genre name for sad or crying movies?
There is such a genre: it is called a drama. The widely used symbol is a laughing face and a weeping face:
Since 100% sad movies have very limited appeal, there is little popular demand for them. An audience which is made to feel a progression of happiness and sadness covers far more emotional ground and is a more fulfilling experience—in some ways paralleling a good horror movie which sequences between high tension and total relaxation several times.
A drama film is a film genre that depends mostly on in-depth development of realistic characters dealing with emotional themes. ...
At the center of a drama is usually ... characters who are in conflict... and touch on the deepest emotions of normal people. Dramas often, but not always, have tragic or at least painful resolutions and concern the survival of some tragic crisis....
Some of the greatest screen performances come from dramas, as there is ample opportunity for actors to stretch into a role that most other genres cannot afford.
There is no such movies like sad or crying movie, all genre can have sad or crying parts. It depends upon how much sad does that movie can make you feel. Although drama movie generally gives a sad feeling as they are not made (not in all case) to make people happy.
If one want to give a name to sad or crying genre - it can't be done, because both of the feelings are different many times. For example a romantic comedy can also make you cry. While black comedy is always sad, it is always what you feel about it.