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A kid is 4 years and 7 months old. Is he in his 4th year or 5th year?

Counting from the onset of birth 0 - 1 - 2 - 3 - 4 - 5.....

Between 0 and 1, one is younger than 1 year and I would count 0th year of your age. Between 1 and 2, you turned 1 and growing towards 2 years. One is yet to turn 2 hence I suspect we say one is still in his/her 1st year.

Mathematically or otherwise, we never say a kid is 2 years, even if he/she is 1 year 10months

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4

By convention, English uses first (not zeroth) to refer to the initial occurrence.

first ordinal number Coming before all others in time or order; earliest; 1st. ‘It is hoped to have everything in order for the first performance at the end of November.’ - ODO

So the period between birth and their 1-year-old birthday would be their first year of life, so the 4 years and 7 months old kid from your example would be in their fifth year of life (or, at least, of life after birth).

Note that this assumes a 'western' system of age reckoning. Other systems of age reckoning exist, for which one's 'nth' year might be different. See below, for example (emphasis, mine)

East Asian age reckoning is a concept and practice that originated in China and is widely used by other cultures in East Asia. People begin life at the age of one (instead of "zero"), and on New Year's Day, one year is added to their age. Since age is incremented at the beginning of lichun, which is the first of the twenty-three solar terms, rather than on a birthday, people may be one or two years older in Asian reckoning than in the western age system. - wikipedia

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