Are you sure you didn't sign any NDA, preventing you from working on ideas you might get during interview? Google used to have pretty water-tight policy on outflow of ideas (but inflow was encouraged) during interviews.
Here's the thing: the NDA you sign when you go onsite covers the company's intellectual property. It might cover the interview questions they ask you, but if they're not paying you for the work you do as part of a take-home interview question, and if you're doing it on your own hardware - it's not theirs, it's yours, and they have absolutely no say in what you do with that work afterwards.
Moreover: if you are subsequently hired by the company, they still don't own the work you did on your own time and hardware. (However: if you continue to work on it while employed by them, they might then own that follow-up work. Check your employment contract carefully, and if you're in a position to insist that employers do not own your off-hours work, insist away.)
Now, if you're currently at another company, and you complete the interview task on their hardware, and especially if you make the mistake of doing that work during company work hours...well, your current employer might then own your interview work. This is a matter usually covered by your employment contract (and another excellent reason to read those carefully).
This applies to the design prompt specifically, which is an early-stage take-home thing. I don't recall whether google had an NDA for the onsite interview portion itself, but I think their badge signin thing might have one baked in, and other companies of equivalent size had paper NDAs as part of checkin. Presumably they only care about you stealing the cool stuff you hear about, and you wouldn't be in danger for running with their (intentionally generic) interview prompts on your own.
Company Tried to Patent My Work After a Job Interview
I never had to provide an actual signature, aside from a verbal agreement with the recruiter to not share interview questions.