I bought a "Great Neck" Metric tape wrench set, but the wrench does not
grip a 6mm tap securely. The taps all have square ends, but there does
not seem to be anything in the wrench that is designed to mate with that
square end.

I think I see the problem. Do you see where is says "Great Neck" on the
tool or the package? Find one that does not say that and it will be a
better quality.