Although I am not a native English speaker I would say there can be some subtle differences between these two expressions.
1.So far this morning: I would get an impression like this: So far, this morning is fine (or perhaps has been fine). It implies that things could change at any given moment. For me, it comes quite near to the expression “until now”.
2- Since this morning: Since this morning I have been studying English. It implies, in my opinion, that I started studying in the morning and I am still studying now – without any interruptions.
Maybe we can use these two expressions interchangeably and every native English speaker would understand what we are going to express. However, in my mother language we have quite similar expressions with the very same sentence structure and there is a little bit of a difference between them. Unfortunately, I cannot say more than to mention that it is always depending on the context these expressions are used in.