A cute character regularly reminding people of their home to take advantage of this tax makes more sense to me.
Mascot characters like Kumamon, which were created after 2008 but before the furusato nozei system really started attracting attention, were specifically created to attract tourism, not to attract donations.
So even though it might seem that the timing would match up, it's not really true that there is a connection.
Furthermore, when municipalities started trying to compete for donations, they didn't use mascots, they just started bribing people with return gifts. Maybe in theory municipalities could have used mascots to try to attract donations, but in reality they don't seem to have done that.