No, Hanna Montana linux is actually a distro, it's a "starting point", as in, a live CD that comes with certain software installed for you. In this case Hanna Montana Linux is essentially Ubuntu but with a different default theme, and yes, that qualifies as a different distro.

This is why the distro terminology is unoptimal. If I were to pick Ubuntu and install that theme, then I would still be on Ubuntu, but if I just took the HML live CD and go from there, then I would have essentially the same system, but run HML. (I don't actually run that, it's a joke, but it does actually exist.)

So there's the problem with distros, they are nothing more than your "starting point", what you ended up at might be widely different from that starting point, completely reconfigured. Like this, my current setup as I'm typing this on a relatively old notebook. This calls itself "mint" but it's as far removed from mint as it is from debian or ubuntu. All three using the same package management system. In fact, if I were to edit some simple text files on my system while retaining the overall functionality called the release files which contain some info about the distro then screenfetch would no longer call it Mint. As you can clearly see my current setup looks nothing like any setup Mint offers, I've reconfigured significant parts of the interface as well as the filesystem. But because it started as a Mint live CD my distro is still "Mint", if I were to install Pacman and get my stuff from the Arch repos from this point on it would still call itself "Mint" and that's the problem with "distros" in terms of terminology.

"Distributions" are not operating systems, they are "software distributions" and every GNU/Linux distro is ultimately built around more or less the same OS at its core. They're just different default configurations and default installations of programs if anything else, and any setting can of course be changed.