When you first register and activate your iPod shuffle, iTunes will give you a chance to pick a name for the shuffle (an especially useful feature if you have multiple iPod shuffle plugged into your Mac or PC at any given point!) This is the name you'll see in the iTunes Sidebar, on the left, where the list of currently plugged in iPod-iPhone-iPad is displayed, as shown on the left screenshot. The iPod shuffle name may also appear in Windows Explorer (Windows 7 / Vista / XP), or in the finder's side bar on Mac OS X - though you may also see a generic name in Explorer / Finder like "Secure Digital Storage Device".
Change iPod shuffle name in iTunes and Windows Explorer / Mac Finder
When your iPod shuffle is plugged into your computer and recognized by iTunes (i.e., showing on the left), select it with a click. Then, click once more on one of the letters of its current name, and it will become editable text - with that approach, start typing the new name and hit Enter / Return to accept it. As soon as you do, iTunes will update the iPod's name on the iPod itself, which means that the new name will be reflected whenever you plugin that shuffle into another computer.
Tip: double-clicking on the current iPod shuffle name will make it editable text without selecting anything; on Windows, press F2 on your keyboard while the iPod shuffle is the currently selected item in the iTunes Sidebar, and you'll be able to type a new name over (and hit Enter to save and apply that new name).
Note: renaming your iPod shuffle will not affect sync, playback rights, no content is removed, or anything else associated with the iTunes Store or Apple ID. Behind the scenes, iTunes keeps track of your iPods using an internal ID: the changeable name you pick for your iPod shuffle is just a convenience, and kind of nickname designed for human consumption only!
So, can you have two different iPods (shuffle or otherwise) with the same name? The answer is "Yes", because these are not used under the hood. But they may become hard to distinguish, especially iPods of same model and version (two 4th generation iPod shuffles will have the exact same icon in iTunes).
Once you've renamed your iPod shuffle, manually eject it from iTunes and you are ready to go!