Which country is the center of Earth?

The simple answer is no one owns or possesses the center of the Earth.

The UN has operational guidelines on these matters of non-recognized territories. Typically, coastal waters are defined as 20 miles out from mean tideline. Territory air rights extend to low orbit, although some nations reject any stationary or near stationary objects above them. Underground is less obvious.

Right now humans (mines for all real purposes) can only extend about a mile down. Small shafts can extend down about three miles. Given the diameter of the Earth (just under 4,000 miles) this isn’t very far underground.

There are two working examples that would likely be used should the matter come up. Antarctica and the International Space Station. In Antarctica, a nation comes along and sets up a base. That base plus an operations area around the base (typically 20–100 miles depending on the declared research and agreements from adjacent bases). There are many exceptions to these that were developed by agreements, so imagine broad guidelines here.

In the case of the International Space Station each module operates like an embassy of the sponsoring nation. All space around it is international, and crimes or disputes depend on who and where it occured.

So, back to the question. In order for a nation to claim the center of the Earth as territory, a human-operated base would need to be set up within a short distance of it. Likely, in order for the UN to accept this claim that nation would need to allow members of UN nations to the base periodically to conduct their own research.