I have heard that languages like Spanish, Portuguese, Italian, and French are classified as Romance languages. Languages like Dutch, German, and English are classified as Germanic languages. All of the Scandinavian languages except for Finnish are classified as Germanic languages. The Finnish language is classified as a Uralic language. So, what is the historical origin of this language classification?
"Romance" is based on the historically-known fact that the languages descended from the language of the Romans, who spoke Latin. The classification "Germanic" is also ancient, dating back to Graeco-Roman times: see here for an overview of Germania in the BC and later years. Historical linguistic classification in that era was not sophisticated: it was simply known that there were such people (and there wasn't a clear distinction between "Germans" and "Dutch", but there was some recognition of tribes within the Germani). There was some understanding that Gauls and Germani spoke a different language. The contemporary name derives from the Latin name (Tacitus wrote a book Germania, which may reveal more details of the origin of the term, but it is unclear where the term came from). The contemporary unity of the people and languages is thus preservation of older knowledge.
Language similarities resulting from Uralic unity was discovered at the end of the 17th century, but the specific details took a few hundred more years to hammer out. There is an article by Bo Wickman in Sinor (ed) The Uralic Languages which traces the history of the study. However, it is not clear when the term "Uralic" was first used (the alternative term, Finno-Ugric, is now taken to refer to a sub-branch of Uralic). The terms apparently dates from the 19th century, and refers to the presumed homeland of the original population. Of the three, families, the discovery of "Uralic" is the one discovery, whereas "Romance" and "Germanic" is based on preserved historical knowledge.