Yep, same root.

Main Entry: mu·lat·to
Pronunciation: m&-'la-(")tO, mu-, myu-, -'lä-
Function: noun
Inflected Form(s): plural -toes or -tos
Etymology: Spanish mulato, from mulo mule, from Latin mulus
1 : the first-generation offspring of a black person and a white person
2 : a person of mixed white and black ancestry

http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=mulatto

Main Entry: 1mule
Pronunciation: 'myü(&)l
Function: noun
Etymology: Middle English, from Old French mul, from Latin mulus
1 a : a hybrid between a horse and a donkey; especially : the offspring of a male donkey and a mare b : a self-sterile plant whether hybrid or not c : a usually sterile hybrid
2 : a very stubborn person
3 : a machine for simultaneously drawing and twisting fiber into yarn or thread and winding it into cops
4 slang : a person who smuggles or delivers illicit drugs

If you don't know it, can I suggest you try onelook.com for standard queries? It's a good starting point.