an indication that an on-device password manager is being used. Banks might reasonably object to that.
Banks might object to the use of password managers. They cannot do so "reasonably", as there are no reasonable objections to them, from the banks' position. The attack tree with a password manager is a considerable improvement for the defender over the attack tree without one.
Password managers introduce new failure modes, but nearly all of them affect the user, not the resource. Those that do affect the resource are highly unlikely and thus have a low relative risk.