They could probably crank up an Uber-style service quicker than anyone else, do it 100% internally, and be instantly integrated into Alexa and whatever else they have going on.
Then the service speeds on Uber's AWS instances start to slowly drop...
Uber has around 4 million registered drivers and millions of daily users and a huge team of engineers working on these problems. With Lyft and Uber in this space it makes no sense for amazon to try and compete with them.
Uber is trying to turn itself into a logistics business and Amazon is investing a ton into making 1 hour delivery work because perishable goods are a huge slice of commerce and a big growth opportunity for them.
Making a copy of Uber doesn't require Amazon to buy 4 million cars. They just need to launch a website and an app. Like another poster said above, Uber drivers aren't loyal to Uber and will swap to Lyft wherever and whenever. A third company doesn't cause any friction at all.
Opening up a few hundred stores is an expensive, time consuming endeavor. In contrast, creating a ride sharing app and promoting it a bit seems like something that a large tech company could pull off much easier.
Groceries, while being notoriously low margin, are the second largest retail business in the US (after gas stations). Launching a new grocery brand, especially one with Whole Paycheck's great brand reputation isn't cheap or easy. See Tesco's Fresh & Easy debacle for an example.