If you are curious as to the specific hallmarks of what defines a Germanic language as opposed to a Romance, Celtic, and so on, you may want to take a look at r/linguistics. The users there, even if the sub is a lot slower than here, are much more qualified to get into the nitty gritty of why English is Germanic.

However I can answer some other parts of your question. The modern state of Germany quite simply didn't exist up until relatively recently, and "German" speakers never conquered England. indeed modern German is not all that much older than modern English, and that's without delvijg into the rabbit hole of German dialects.

English is Germanic but not German if that makes sense. Specifically English belongs to the west Germanic language grouping, along with Frisian and Dutch. The English we speak today goes back ultimately to the Old English of the Anglo-Saxons who migrated to what is today England in the 5th-7th centuries. Without specific training, this variant of the language is nearly unintelligible to a modern English speaker. You might be able to pick out some words in isolation, but as a whole it would sound extremely unfamiliar. Some words would be very recognizable even today however. Words for family relations, basic colors, common everyday verbs, and so on, have changed only a little since this time.

This language came to the British isles and displaced the Romance/Celtic language of the Romano-British in what is today England (and parts of Scotland). Old English remained the language of England until after the Norman conquest when the sudden infusion of Norman French into the upper echelons of English society helped transform the language into Middle English. That is the English of Chaucer. It is much more intelligible to a modern English speaker, though there are still some quite glaring differences.

If you've got questions on the Anglo-Saxons or the migration era in general, take a look at the FAQ, there's quite a few answers in there about state of Britain at the end of the Roman empire in the west. There are also some answers in there that touch on the relationship between Old English and Old Norse, as well as the change of the English language over time.

Towards the end of the western Roman empire, Britain was inhabited by Celts, and the Germanic tribes lived in northern Europe. After the Romans left Britain in the early 5th century, some Germanic tribes (the Angles, the Saxons, and the Jutes) migrated from what is now Denmark and northern Germany to what is now England. They spoke old English, a West Germanic language.

Starting in the 8th century, Norse tribes speaking North Germanic languages (mainly Danes, but also Norwegians) began raiding in Britain, later mounting full scale invasions (most famous are the Great Danish Army of the late 9th century, and the Kings of Denmark in the early 11th century). Over this time, Danes settled in England, particularly the North and the East, and their language had an impact on the original West Germanic language.

The Normans contributed large numbers of words to English, but English can still be considered Germanic for several reasons.

Firstly, languages are classed based on their evolution and grammar, and English's is Germanic.

Secondly, although there are now many words of French origin in the English language, core words are almost all Germanic. I think it's something like 97% of the 100 most common, and 80% of the five hundred most common words in English are Germanic. A good example of this: you could have an English sentence composed entirely of Germanic words, but you couldn't have an English sentence composed entirely of non-Germanic words.