Yes, they do track multiple things... But only one thing per item type; Kills for guns and MVPs for music kits.
Edit: Updated for the music kit update (which was ages ago by gamer term; around 3 months at the time of writing)
StatTrak sadly, only tracks player kills done with the gun (or MVPs earned for Music Kits) that it is attached to...
Uninspiring/creative.. But that's the only thing that StatTrak item can do for now; track one stat per item type, as opposed to DoTA 2 whose Inscribed Gems can count anything (including hero spell casts) and times a particular thing happened.
Picking up a StatTrak gun that's not yours (ie. Dropped in Casual upon enemy's death) will not add kills to it either.
Even if it tracks anything else, it is most likely undocumented or even released (in the code) at all... Until the time it is meant to be released or announced.
Also, Valve hasn't announced anything about tracking other stats... As for the current operation; Operation Breakout - The missions only track kills with a certain weapon (skin/whether it has StatTrak or not does not matter) and wins (outcomes of a competitive match).
All StatTrak guns have currently is value, as they're rare to get from crates (and impossible from what I've heard so far from item drops)...
The most you can really do with a StatTrak item is for bragging rights or a show off on experience/skill with the number of kills done with the gun or MVPs won with the item equipped (it counts)... Although it can be inaccurate and easily exploited. (Try killing AFK people on an AFK server to get kill counts onto the gun to quickly add kill counts to the gun)
And as for StatTrak Music Kits... They only cost more to buy, as compared to their non-Stat-Trak counterparts.