First of all, I'm not an expert on Finland nor do I know much at all about ancient Finland. However, I do know about this meme and I can tell you where it comes from - but I fully expect this post may be deleted, even if it answers your question, because it's not really a history-related answer.

The "Ancient Finnish Empire" meme is a joke popularized on 4chan's /int/ (International) and /his/ (History & Humanities) boards. It's somewhat related to the "fingols" meme (which afaik orginated on /int/ in threads where people would ask why Finns look so much like Asians), as well as the "Finno-Korean Hyperwar" meme, a fake war between huge, ancient, technologically advanced (fictional) Finnish and Korean empires often discussed on /his/ in jest as if it were a reality.

The origin, as far as I can tell, of these memes are maps depicting large Finnish and Korean empires that supposedly existed in ancient times, as a joke. This Finnish one depicts various famous European cities as having been originally founded by Finns, as well as the likes of ancient Greece and pre-Roman Britain having Finnish founders and precursors. It's pretty safe to say this is an exaggeration of any potential greater Finno-Urgic ancient empire for the sake of humor.

There is a running joke on /int/ that nationalists often harken back to their country's "golden age" when espousing their country's glory - Serbians to the Serbian Empire of the 14th Century, Greeks to the Byzantine Empire, Armenians to the ancient Kingdom of Armenia. The memes about a greater Korean or Finnish empire are a parody of this, a spoof of nationalist romanticism taken to a silly extreme. I would advise you don't take /r/fingols or anything like it as a serious historical source. It's more of a joke along the lines of /r/badhistory.