If aluminium hydride, $\ce{AlH3}$, is called "hydride" because it contains elemental hydrogen and sodium chloride, $\ce{NaCl}$, is called "chloride", and calcium carbide $\ce{CaC2}$ as another example, then why is methane not called carbon hydride or carbon tetrahydride.
Another question why hydrogen halides, e.g. $\ce{HCl}$, $\ce{HBr}$, $\ce{HF}$, are called like this and not halogen hydrides.
The name methylene has existed in English (and French) for hundreds of years, and is based on the far older English mead and Greek root μέθυ (methy) for alcoholic beverage. In a similar fashion, ethyl has become part of the language from aether, or αἰθήρ.
So, $\ce{CH4}$ might be called carbon tetrahydride, as $\ce{SiH4}$, silane, is silicon tetrahydride, and $\ce{H2O}$ is dihydrogen oxide. More accurately, though, as Oscar Lanzi points out, it should be hydrogen carbide, because the hydrogen is more metallic, i.e, more generous with its electron, as in the case of $\ce{Be2C}$, beryllium carbide.
However, traditional names are more commonly used, as you can check on Google's Ngram Viewer for water. Personally, I'd rather that helium was helion, named as a metal by Pierre-Jules-Cesar Janssen, but c'est la vie
As DrMoishe Pippik describes in his answer, historical usage has rendered "methane" a sufficiently recognizable name to be promoted to technical usage. But beyond that, a look at electronegativities indicates carbon is more electronegative than hydrogen. Thus methane would be more accurately rendered as a hydrogen carbide than as a carbon hydride.
Metal carbides that give methane upon reaction with water or steam, such as $\ce{Be2C}$ or $\ce{Al4C3}$, are often called "methanides", which goes along with methane being a (covalent) carbide.