The modern word family has broadened to the extent that we can talk about, for instance, a family of languages, without suggesting close links. But originally it had nothing to do with relatives at all. It sprang from the Latin famulus , servant, and broadened into familia , group of household servants.
Temptation comes from the Latin temptare , handle, from tendere , stretch.
I can't give you a definitive answer about the switch in meaning from test to allure , but I believe it would have been an easy one. The test of resisting that which was alluring became the thing itself - the temptation .