The Finnish sentence should be "Tämä on joko suomea tai viroa."
Should this not be the mutually exclusive vai? If it is one it cannot be the other.
Vai is used only in questions.
It's not quite the distinction of OR vs XOR. If it's possible to use "either ... or ..." in English, it's tai. Vai only appears in questions asking the listener to choose an option (I think also in imbedded questions, but I'm not sure).
"either...or" should be translated "joko...tai"
Why partitiivi here?
The "tämä" referred to is only a part of either language, not the whole Finnish or Estonian language.
Why is sekä... että wrong?
That would be both Finnish and Estonian.
Why is the ei...eikä construction wrong here?
Because that's pretty much the opposite. This is neither Finnish nor Estonian.