You are still legally allowed to 'check yourself out' which would be considered 'against medical advice', and god knows what you'd have to sign, but the point is that's not typically the way it works. You are under their care until a responsible party comes in to retrieve you.
Whether this is actually the case is a different matter, but that veiled threat should be enough for most.
(I suppose they might physically restrain you if they have reason to think that you aren't mentally competent at the moment...)
Nitpick: false imprisonment, not kidnapping (kidnapping requires movement).
It's not a common enough use case for Uber/Lyft's consumer facing services, so I don't think it's wise for them to get into it; they're better going after UberEATS/etc opportunities. The idea would be ridesharing for businesses who have more strict requirements, kind of like Box vs Dropbox.
FWIW, you still deal with the difficult people operations aspect of ridesharing in that case (drivers are hard to recruit and manage, and are city specific), so I'd probably go after outsourcing the logistics platform for businesses with existing delivery mechanisms. E.g. bring UPS's 'no left turns' technology to companies that can't develop it internally.