Origins
The moon was humanity's first ruler, and the word measure remembers this. It comes from Latin mēnsūra, from mētīrī ('to measure'), from Proto-Indo-European *meh₁- — 'to measure'. The same root produced moon and month, because the lunar cycle was the original unit of time.
The PIE root *meh₁- branched into two great language families. Through Latin mētīrī it produced measure, dimension (measuring across), immense (beyond measure), and commensurate (measured together). Through Greek metron it produced meter, geometry (earth-measuring), symmetry (measured together), and diameter (measuring through).
The Germanic branch preserved the root in its most elemental form. Moon comes from Proto-Germanic *mēnōn, the measurer. Month comes from *mēnōþs, a moon-measurement. Before clocks, before calendars, the moon measured time by waxing and waning.
French Influence
Old French mesure carried a secondary meaning that English inherited: moderation. To act 'beyond measure' was to act without restraint. A measured response is a moderate one. This sense connects measurement to wisdom — knowing limits requires first knowing quantities.
The political sense — 'a measure passed by Parliament' — developed in the 14th century, from the idea of a calculated, weighed action. Every government measure is, etymologically, something that has been carefully measured out.