We analyze the precise data obtained by the CMD-3 experiment on the e(+)e(-) annihilation into two charged kaons in the vicinity of the peak. A perfect fit is obtained only if a pole on the real axis below the reaction threshold is assumed. This can be interpreted as proof of the existence of the 2p state of kaonium, a compound of K+ and K-. The BABAR Collaboration data on the same process supports this conclusion and, in addition, points to the strong interaction as a dominant source of the binding energy. The possibility of discovering 2p kaonium in the e(+) e(-) -> pi(+)pi(-) process is discussed. The 2p state of K-0-onium is indicated on the basis of the CMD-3 e(+)e(-)-> K-S(0) K-L(0) data.