There are really two issues here, but the answer to both of them is "no." You can use other software options on your iPod, instead of iTunes, to manage your content on the iPod. On the other hand, you can also use your iTunes on other media in addition to your iPod (if you even have an iPod).
For this first issue, there are other software options, including freeware (software applications you can download and use for free), which you can use on your iPod. Different software provides different functions, so first figure out what you need. Some of these other iPod (really, mp3 player) software selections only change the interface while others include the actual audio functionality. You'll want to investigate your options carefully to make sure you're getting all the functionality you need and want (such as managing podcast subscriptions, creating play lists, etc.), as well as to ensure that the new software works with your mp3's specifications. Pay attention to the installation directions; these will help clarify if you can use any specific mp3 file management software on your specific mp3 device.
Now to the other issue: It really is more accurate to talk about what you can use on your mp3 player; iPod is just one example. A number of manufacturers make their own versions of an mp3 player, not just Apple. iTunes, however, is an Apple service. Having said that, Apple wants as many people to use iTunes as possible. For this to happen, Apple had to make iTunes usable on more platforms than just their own. So you can download the iTunes app (and thus songs off iTunes) to any number of mp3 players. In fact, you don't even need an mp3 player to use iTunes; you can use iTunes directly on your computer, Windows or Mac.