The device was class II, but it was powered by a commercial (medical grade) power adapter on which the connection to earth was used for the EMC screen.
I would be suspicious in this case. In 25 years I have seen a couple of cases of Class II equipment with a functional earth, which is then confirmed to have 4kV/8mm spacing from mains parts as required in IEC 60601-1. But these had custom designed power supplies, not off the shelf. I never saw an off the shelf Class II "Medical grade" power supply with a functional earth. I expect it not to be commercially viable because the application is rare, and the potential for confusion is also high.
For the original question, I think it is not an issue in terms of compatibility with IEC 60601-1.
A device labelled as Class I (i.e. no double box symbol) but has all internal earth connections double insulated from mains parts (8mm/4kV) and only used for functional purpose still complies with IEC 60601-1. While unusual, there is no point of non-compliance in the standard, and no safety issues.
Also, this is the best option. The potential confusion from labelling a 3 pin device as Class II is not worth it. If the earth is used only for EMC, then it still needs to be connected to earth anyway to comply with regulations. The user does not know the internal construction so it makes no difference to them if the internal earthing is functional or protective.