That image is more than a little silly, I really don't know where to begin. If the creator of the image wants to make Finland the "legitimate" successor to the Roman state (whatever that means exactly) simply on the basis of descent from the Russian Empire, then why wouldn't Poland also be the legitimate successor to the Roman state? Both countries were only made independent at the end of WWI, within months of each other (both, mind you, after the end of the Russian Empire in 1917, so no the Finns were not free of "Soviet taint"). And how do the Ukraine and the Baltic states somehow "lose legitimacy" by being not "free of Soviet taint" (wow, aggressive nationalism much)? That's rather a silly thing to say. While it's on my mind, Finland was only incorporated into the Russian Empire in 1809 as the Duchy of Finland, so I don't see why they should be considered an integral part of the Russian Empire. Sillier still is what constitutes "legitimacy" to this person. What the fuck does it mean to be a "legitimate successor" to a state, especially one that hasn't existed for millennia? To this person it seems the criteria are more or less that whatever states he feels like should be declared illegitimate are illegitimate. What, precisely, makes the Russian Empire a legitimate successor to the Roman state but the Ottomans are not? Neither of them are Roman, neither of their rulers thought of themselves as Roman. Sure, the tsars married into the imperial family, so did lots of kingdoms. The Russian Empire controlled no major part of the former Roman state and didn't really have much of a cultural connection to them. What makes them more legitimate in calling Moscow the Third Rome then the Ottomans, who controlled most of the old territory of the Eastern Roman state, including the capital itself, and many of whose people still thought of themselves as Roman until quite late? I mean I think the most laughable thing here is that the Western Roman Empire is just declared offhand as illegitimate. What? What?! The nationalism is strong and arbitrary in this one

The fact that someone is interested claiming a Finnish inheritance to the Roman Empire is actually far more interesting than determining whether the claim is a "legitimate" one.

With regards to claims and legitimacy, understand that legitimacy is a negotiated communication of authority with those involved. If the restriction of this circle of communication is to the creator of this graphic and you, and you agree, then it's legitimate. If your circle of communication increases to the opinions of scholars and representatives of modern states, then no.