Same here. In the States I mostly use Lyft, but internationally it's easier to just use the Uber app instead of first figuring out which other options are available. In many cases installing the local apps doesn't even work because they are limited to their own local app stores.

One of the next steps for those companies is probably to get roaming agreements with each other: When I'm in a country where, e.g., Uber doesn't operate I'd still like to be able to use the Uber app to order cars from whatever local companies Uber has an agreement with.