That's really just a Finnish thing.

But to start at the beginning, basically all Swedish kings (and viceroys, Sten Sture shouldn't be forgotten) that had the courage to die leading their men in battle were exalted to the status of hero-kings afterwards, even if their wars were otherwise ruinous to the country (read: Karl XII). Gustav II Adolf's wars (the 30 Years War) were not; that war leaving Swedish with Ingria, Estonia, parts of Livonia, and parts of Northern Germany, and material treasures from sacking Prague and other continental cities. It marked the beginning of the Swedish 'Era of Great Power' (Stormaktstiden).

So the king may have died in battle in a rather ignominious way; it was a foggy day at Lützen, and being of bad eyesight (he would've needed glasses) he mistakenly rode into enemy ranks. But the fact that he did die in battle, and the war was nevertheless won and brought large increases in wealth and territory to Sweden made him instantly a great hero king. Livrustkammaren in Stockholm still has his bullet-riddled clothes on proud display to this day; even his horse was taxidermied and put on display! Nor did it hurt he was the last king of the Vasa line (who'd been built up by skillful propaganda since its founder Gustav I) and that the war had a selfless justification as being about defending the Protestant faith. One could compare to his successor Karl X; who, although highly regarded, never reached quite the same legendary status, even though the conquests made by Karl X were more significant historically as they ended up being permanent (i.e. Gotland, Skåne, Blekinge, Halland and Bohuslän remain Swedish to this day)

Anyway, so in the rising nationalism of the 19th century, Gustav II Adolf's death began to be celebrated in Sweden as 'Gustav Adolf's Day' (Gustavadolfsdagen), on November 6th, and could to some extent be considered Sweden's first patriotic/national holiday. (November 30th, the day of Karl XII's death was also commemorated to a lesser extent; although since the 1990s it has been appropriated by the extreme right) The current national holiday (June 6) has only been such since 1983, prior to which it was the 'day of the Swedish flag', established formally in 1916, although it'd been celebrated since the 1890s as a 'day of patriotic memory' on the initiative of Artur Hazelius (creator of Skansen and the Nordic Museum in Stockholm). With the rise in the 20th century of june 6th, and relative decline of nationalism, Gustav Adolf's day faded somewhat, and is today reduced to flying flags and eating effigies of Gustav.

In any case, when the Swedish People's Party (Svenska Folkpartiet, SFP), representing the Swedish-speaking minority in Finland wanted a day to celebrate their heritage, Gustav Adolf's Day was the best option. June 6th did not have formal status yet. Also, it is in commemoration of the adoption of Gustav Vasa's coronation and the adoption of Sweden's Constitution of 1809, both of which happened on that day. But only the former event has any relevance to Finland, as it just come under Russian rule (indeed the constitution was a result of the political fallout of that loss)

Another option would've been Runeberg Day (Feb 5th), the day of his birth, but although Runeberg and Fänrik Ståls Sägner are well-known in Sweden, he does not have the same status. Nor would he - Runeberg was important in helping form the emerging Finnish national identity in the 19th century. He didn't and couldn't play the same role in Sweden. So Runeberg day is not celebrated in Sweden, and would not have worked as well in SFP's goal of a day promoting Finnish and Swedish mutual heiritage.

But Gustav II Adolf's day was a patriotic day for which Finland was every bit as much a part as Sweden-proper. The Finnish hakkapeliitat are almost legendary and there were important Finnish generals like Torsten Stålhandske.

So when SFP wanted to create a day for the Swedish-speaking Finns back in 1908, they went with Gustav Adolf's day. Initially the day was focused more on G2A himself, but like in Sweden, the patriotic cult of old kings cooled a lot after WWI and further after WWII, hence it is now almost more an excuse to eat pastries than anything in Sweden (and perhaps a bit like Runeberg day in that sense). Finland would soon gain independence and it's own independence day, which would be the natural focus as a patriotic day. So Svenska Dagen came in Finland to be less about Gustav II Adolf and patriotism and more about the Swedish language.

(Which, IMHO wouldn't be a bad idea to adopt for the day in Sweden itself)