For the driver this would just appear like their current destination suddenly changing, which I guess you have no option to deny except for asking them to leave your car and canceling the trip entirely.
Being able to make major trip changes mid-ride is pretty crappy for the driver. I'm also surprised the multiple stop feature isn't just for simply letting someone in/out of your ride. But apparently Lyft actually encourages you to use it to stop at the store. [0]
> Whether you’re picking up a friend or a bottle of vino, just add your stop into the app and your route will instantly update — making it a seamless experience for you and your driver.
The person in the article was driving for Uber. The verbiage on Uber's site for this feature is more focused on passenger pickup/dropoff, but also doesn't make it clear whether you're allowed to spend time visiting a store. I found a rideguru post indicating you're limited to 3 minutes at a stop, but can't find this officially.
[0] https://blog.lyft.com/posts/add-a-stop
[1] https://ride.guru/lounge/p/whats-the-2-stop-rule-on-uber-is-...
Edit, found this on an Uber blog post. Do riders really know, Uber? Do they?
> Riders know that each stop should be less than 3 minutes, so you can get back on the road as soon as possible.