These two sentences are funny, as in self depreciating humor.
1 Our family is not so good that it is degenerating.
Means: Our family is not wealthy, or powerful enough to be marrying our cousins or other close relatives to maintain wealth or power.
2 As yet no physiognomy in our family is coming out in edition after edition on increasingly thinner paper.
Means: From generation to generation the shape of our noses, eyes or smiles are different.
The second sentence is re-enforcing the first. Saying that as well as the family not being wealthy or powerful, it is also doesn't have strong physical characteristics.