Is there any data available on how many people use an alternative client in contrast to the Satoshi client? Alternatively, is there any data on how many times such clients have been downloaded?
MultiBit is a thin-client based on the BitCoinJ library. It is written in Java and comes with installer packages for Mac, Windows and Linux. It is intended to be used by the general public who don't have a great deal of knowledge about Bitcoin and just want to send and receive payments in the simplest manner possible.
The project owner on the repo has the info on the number of downloads but unless that is shared by that person I don't think that information can be obtained.
There is a "traffic" statistic that can be seen on GitHub. For example:
BIP 14 addresses a method for a client to identify itself:
Exposing this information is harmful though when there is the desire for security through obscurity so the Bitcoin.org client now reports "Satoshi", apparently.
You can certainly find statistics on what version of the protocol are currently most commonly reported, and since certain clients are/were more prevalent in those versions of the protocol you could potentially extrapolate some data from that, but to my knowledge, the actual client application name is not reported and so this data is not public information.
Those running the project could certainly provide download statistics but that does not necessarily reflect how many people are actually using the client, only how many have "tried it out."