Legally or morally?
Why the overlap, did you screw up? If it were me, I’d have engineered a free holiday in between roles. 😁
It sounds to me like they’re offering to let you leave early on the proviso that, well, as you say, you’ll field occasional questions like “what’s that password?” I would be astonished if they were offering you full pay whilst working for someone else, the chances of you drawing two salaries are surely vanishingly small. Assuming even that your line manager has the authority to override your contract in the first place.
In practice, if you were to say “no thanks” and just leave, would they pursue it? Probably not, but burning bridges is rarely a good idea.