When two languages have a common ancestor, we say that they are 'genetically' related. This has nothing to do with genetics in the biological sense, and it doesn't imply that speakers of genetically related languages are themselves genetically related. But languages share ancestors like organisms share ancestors, and so genetic relatedness is the metaphor historical linguists use.
Unlike with organisms, genetic relatedness rarely tells us everything we need to know about a language. If you said that English is a West Germanic language or an Indo-European language, you'd be saying things that are true, but that are just not the whole picture.
English has plenty of non-Indo-European-derived words, including 'person' from Etruscan φersu and 'shark' from Yucatec Mayan xoc (which entered English around 1585). The Germanic languages themselves appear to show the presence of a non-Indo-European substrate language - that is to say, a non-Indo-European language that crossed with Indo-European at some point in European prehistory, affecting the structure and phonology of proto-Germanic. While proto-Germanic is clearly an Indo-European language, it shows the presence of some obviously non-Indo-European languages.
English's structures are not wholly West Germanic, either - the idea of forming a question with the verb 'to do' as an auxiliary, e.g., 'do you like linguistics?', apparently derives from a Brythonic precedent. North Germanic languages contributed so much to English after the Norse invasions/migrations that even the verb 'to be' is partly Norse; 'he is' is entirely Anglo-Saxon, while 'they are' is entirely Norse. Words like 'egg', 'sky', and 'bag' have origins in Old Norse as well.
It's not that English isn't a West Germanic language, but rather that that fact is not the whole picture, and in order to account for the English spoken today, or at any period, we have to look at all the parts. And as ever with human language and culture, those parts come from all over the world, and have origins you may not initially suspect.