Now we understand that Germanic (English, German, Swedish) and Romance (French, Spanish, Italian) are both branches of western Indo-European languages and have found thousands of shared roots and cognates between the two groups. But when the Germanic tribes invade Rome, did people know that these two language group (Romances were mostly Latin and Vulgar Latin at that time) are related? What about the Medieval time? Did they finally understood the relationship after the establishment of modern linguistics, or earlier? Thanks for answers!
Edit: By "people" above, I mean scholars and writers that are familiar with linguistics, not regular people.
The first thing to note is that "related" has many different senses, and that the modern linguistic sense of "sharing a common ancestor" still isn't understood by probably the majority of laypeople even today. In my experience the average person doesn't understand what it means when I say two languages are related.
The idea that the European languages are related (again, in the technical linguistic sense) dates back to at least [William Jones](https://en.wikipedia.org/wiki/William_Jones_(philologist) in 1786, who proposed that the similarities between Greek, Latin, Sanskrit are best explained by them having a common root. Of course, people had noted the similarities between the languages before, but the Romans attributed the similarities between Latin and Greek to Rome having been settled from Greece (this idea is visible in the Greco-Roman mythological hero Aenas). For this reason, Jones is often considered the founder of modern historical linguistics.
In terms of Germanic, you might be interested to know that Jacob Grimm, popularly known as one the Brothers Grimm who authored several fairy tales, was the first person to identify a systematic linguistic sound change, which is now called Grimm's Law.