Siblings definately do, cousins don't because they don't share the same mother. They can and do share the same grandmother but that would be apparent.
If the mtDNA was passed on without mutations from the grandmother and also passed on without mutations from the mothers (assuming these cousins are children of sisters), the cousins certainly would have the exact same mtDNA.
Grandma has H and gives H to Daughter1 and Daughter2. Daughter1 has H, married to Husband1, and gives H to daughter1-1 and daughter1-2. Daughter2 has H, married to Husband2, and gives H to daughter2-1 and daughter2-2.
Of course, there are natural mutations to our DNA whenever you make a copy, which means Grandma is not TRULY giving "H" to her children - nor did her mother give her "H." She is giving them a strain that can be traced through her family tree by comparing it to other strains - her piece of millions of strains that, according to Mitochondrial Eve thinking, would eventually ALL trace back to the first "human" woman - the woman that marks the real separation of HUMAN as a unique species apart from others. This Mitochondrial Eve, too, would have prior mtDNA that can be traced back through the previous species, before she became uniquely human, to another mitochondrial landmark. This same mitochondrial landmark individual - or the one before it, or the one before it - also contributed to Mitochondrial Ape. It does NOT mean that Mitochondrial Ape necessarily contributed to Mitochondrial Eve. It means that it's entirely possible Mitochondrial ApeEve contributed to the individuals that went on to later contribute through evolution into Mitochondrial Ape and Mitochondrial Eve.
Which means a wolf and a dog could share strains of mtDNA, but it could be because they both got it from a common ancestor - Mitochondrial DogWolfCoyoteFox, who contributed to individuals who went on to evolve into Dog, Wolf, Coyote, and Fox, and whatever else we might want to chuck in there just for fun - NOT necessarily that Dog got it from Wolf. After all, dogs, wolves, and coyotes are all so closely related they can freely crossbreed and produce fully viable offspring, so it's likely that the same common ancestor contributed to all three individual species.
I hope this all makes sense.