I wish the mods would delete some of these posts arguing that ride-sharing has to be defined as taxi-service. It's a political topic that detracts from the issue of how to use ride-sharing to avoid car ownership.
Obviously a rental car agency is not a taxi service, because drivers rent the cars in the interest of driving themselves primarily (though of course they can drive anyone they want around, even ride-share passengers, for example).
But just because rental car agencies rent out cars to drivers doesn't mean that someone who allows their car to be used by another driver via a ride-share app would be renting out their car. Lending your out and being compensated for the costs is different from renting it out for profit as a business with employees and other overhead costs.
The overall issue with sharing here is that people can share things without doing so for profit the way other businesses do; i.e. because they have the alternative goal of reducing their costs by doing more for themselves.
At the consumer level, reducing your personal costs and liabilities by sharing things makes sense. When you are operating a business, the goal is profit maximization by charging prices the market will bear in a way that brings maximum revenue.
Those are two different objectives, but the consumer objective of reducing cost and liability is no less legitimate than the business objective of increasing revenue by expanding and growing your business.
As a consumer, you don't necessarily want to go into business with everything you do for yourself. E.g. you might be good at fixing spokes and you bought a set of freewheel/cassette tools. You might not want to go into business fixing other people's spokes and wheels, but you could still share your tools and knowledge to help people fix their own wheels. You might not want to open an entire business devoted to renting out tools, but you could still share your tools and people could pay you for borrowing them to compensate you for wear-and-tear and generally share the cost of the tools.
Likewise, you could lend your car out within a ride-sharing network without opening an entire car-rental business. You can also drive for a ride-share app as an independent contractor without that being your primary occupation. If no one was allowed to participate in such sharing networks without being hired by a business with all the expenses and requirements that come with that, then all those things accomplished at a low cost by share-efficiency would be lost to inefficient business models where everyone and their mother is on the payroll so prices are driven sky high and no one's allowed to do anything without someone else's approval.