No. As I understand it the previous system, NemID was actually (co?)designed by the banks so this is what they all use. Likewise MitID is another unholy alliance of Nets (a Danish payment provider) and Danish banks.
Given the Swedish version of it is called BankID I assume the situation is nearly the same in Sweden.
Sweden have one other viable alternative that is Freja ID, it does not have at all the coverage as BankID but it's something.