Can a government ruled by a monarch have legitimacy

As long as there is no revolt against the monarch,  then yes.

A government is considered to has a legitimacy if its acknowledged by the majority of the people and the govenrment conform to the laws that were agreed upon in that society. In this type of monarch, they usually establish a separate parliament that is not influenced by the king or queen to create the legislation

yup! you are correct it is b.