etymology -- have = must?

Why is that in the English language the word ‘have’ can be used in the same sense as ‘must’?

Not “have” but “have to” is synonymous. It carries the meaning, more or less, of “have [the obligation] to.”

Interestingly, the Spanish for “to have,” tener, works in much the same way, when followed by que.

I must wash = I have to wash = Necesito que bañar = Tengo que bañar