They had originally been different groups of people, and did not have 'common ancestors', at least in the most popular hypothesis in latest decades.

As for the 'ethnogenesis' process hypothesis of the Sámi people, please see my summary in How/why were the Sámi people prosecuted in Scandinavia? When did they become separated from other Scandinavians?.

In short, the latest popular hypothetical model supposes:

1. the Sámi people had in fact shared a common ancestor with southerner Scandinavians (Norse people), and the divergence occurred in course of Bronze Age and early Iron Age, mainly based on their living style as hunter-gatherers.
2. They adapted the language later from the diverse farming groups of people of southern Finnish people (to-be).

On the other hand, the identity formation process of the Finnish as coherent ethnic-nation group could be a bit tricky and perhaps wait for the period in the 19th century when the establishment of Grand Duchy of Finland (Suomen suuriruhtinaskunta) stimulated the identity formation of different social groups of people like the officials as well as intellectuals in the Romantic period under the Russian rule (Cf. Kirby 2006, Chap. 3).

You can also check some examples of how the Finnish peoples could be understood from the eyes of neighboring peoples in the Middle Ages and Early Modern Period in the following threads by the comments of /u/Platypuskeeper and /u/SgtBANZAI, though I suppose they primarily appeared rather as diverse local groups like the Tavastians, Savonians, and Karelians throughout the Middle Ages, not under a single 'Finnish' label:

• What was life like in Finland some time before the Swedes arrived in the 1100s?
• What were the Finns doing during the Viking age? What kind of contact did they have with their Nordic neighbors?

Reference:

• Hansen, Lars I. & Bjørnar Olsen. Hunters in Transition: An Outline of Early Sámi History. Leiden: Brill, 2014.
• Kirby, David. A Concise History of Finland. Cambridge: Cambridge UP, 2006.