Little girls can have a bloody discharge from the vagina for a couple
of weeks after birth, but at two months, this is unusual.  Increasing
frequency at two months does not fit a normal pattern at all.

I think a proper evaluation, by a pediatrician who will not tell the
mother to stop breastfeeding (since I cannot even imagine how this
could be related to breastfeeding), is in order.

One possibility, an older sibling put something in the vagina.
Foreign body in the vagina is a common cause of discharge, often bad
smelling, and/or blood.  (Doesn't sound like what you describe).

Jack Newman, MD, FRCPC