No other country (here I am using UN membership as a definition) has official diplomatic ties with this notional micro state. Therefore the answer is, "no, it is not recognized by any other nation"
South Korea in particular does not recognize any sovereignty, so them it is just a claim not worth dealing with. An individual can add any restrictions he wants to people entering his property, but if this republic were to seek to deviate from South Korean law, the cops would get involved.
The situation is mostly likely akin to the "conch republic" of key west. A notional idea at best, but don't try to go against the cops with a blockade.
It is possible that other non-member micro states may choose to recognize this one, but by the common definition of country that the general public would recognize- ie ones generally recognized- the answer is no.