Why is "weight" = 体重 and why is "height" = 身高?
身高 is short for 身體高度

體重 is short for 身體重量

身體 can be shortened to either 身 or 體

Therefore, 身重 and 體高 are also legitimate shortened forms of 'body weight' and 'body height' 

Especially 體高-- it is the proper term for describing the height of non-human object. 

For example: 體高 in "高達體高18米" (Gundam's height is 18M) is the correct term, because Gundam is a machine, not a person

We use 身高 and 體重 far more often than 身重 and 體高 when describing a human's height and weight, eventually, no one use 身重 and 體高 to describe a human's height and weight anymore.