The idea of Finns as east Asian looking comes from hundred year old racial theories, where Finns were seen as "non-white" and "mongoloid" as to prove they are inferior to "white" Europeans. This idea was then propagated by Western Europeans seeking to prove their supermacy. Thea idea of Finns as "mongoloid" partly originated from Finns having a language unrelated to Indo-European. The hundred year old incorrect racial stereotype still exists, and has moved on to be an internet meme joke, contributing to google searches you make.
When it comes to "Asian eyes", or eyes with
epicanthic fold, skin from upper eyelid that covers the inner corner of eye, they are not restricted to Asians. They are a feature which pop up all over the world. For example several African populations have epicanthic folds without being Asian. It is common to children of any race to have varying degree of epicanthic folds as a child, and most lose it when they grow up. Frankly, the reason why some populations in the world have them and some do not is not known, but in Finland they are not common.
If you see what average Finns look like, like average woman (top row, second) here:
Or average Finnish man here:
They look like average Northern Europeans.
Now, Finns do commonly have N1-haplogroup which is common in Northern Eurasia, also with people who "look east Asian". But N1-haplogroup is so old (over 10 000 years) that we have no information what kind of people it originated from. The geographical origin does not tell us much. For example the haplogroup K2 likely originates from Southeast Asia. Descendant of that haplogroup is R-haplogroup, which is the dominant haplogroup in Europe. Still, Europeans are characterized by their lack of epicanthic folds, even though their dominant haplogroup ancestor likely originates from Southeast Asia.
Likewise language isn't that good indicator either. Mexicans, Indians, Iranians, Greek, Irish and Norwegians all speak related Indo-European languages, but their related language is not an indication of how related these people are. For Finns the most related people after Estonians are Swedes instead of Uralic speaking Sami people.