The two are very similar at a surface level, but there are subtle differences when you examine the details.

So, who gets the right? The Mandate of Heaven does not specifically give a right to rule to those of Noble blood. It’s simply reinforces the existing right of he who is already in charge or has taken power (so long as he/she remain a just ruler). The Divine Right of Kings is not so egalitarian. Those who “deserve” to rule are put in that place by god, with the most common interpretation meaning being born of the royal line.

The Mandate of Heaven also reserves a “right to rebellion” for the people. If the current government is unjust, then it is up to the people to rebel and restore rightful and just rule. The Ming Dynasty was founded by one such man, a commoner/peasant who led a rebellion against the Yuan. No such right exists in the Divine Right of Kings. Any attempt to depose, dethrone or restrict his powers runs contrary to the will of god and may constitute a sacrilegious act.

The connection between Divine Right and Mandate of Heaven is not coincidental. But it isn’t an example of cultural/institutional exchange either. Power needs to be justified and believed, and cultures the world over justified their system of power distribution (ie, why kings and nobles have lots of power and peasants have none) through religious or mythical frames of thought. As you’ve mentioned, China has the Mandate of Heaven, and Europe the Divine Right of Kings. But there’s more still. India has the Caste System, which is full of religious meaning. The Muslim world had the Caliphs (the successors to the Prophet). And Japan has an Emperor who was nearly directly implied to be a God or at least semi-divine. And the Egyptians did the same with the Pharos.