Interesting question. The 동 is obviously 同 "same." The 갑 is 甲, the "A" of a sort of Chinese "A B C D ..." ordering series (starts out 갑, 을, 병, 정, ... in Korean) but I don't see why 同甲 should mean "same age." All I can find for 내기 is as suffix for "ready-made articles, goods in stock." The only 환갑 I find is "full 60 years of age" and 환갑년 "one's 61st birthday."