Israel is not a staunch ally of the U.S. They have their own interests, and sometimes they coincide with ours.
The U.S. has no such formal relationship with Israel except in one very limited circumstance, and that is as much for the protection of Egypt as it is of Israel.
For a new U.S. president to invite a foreign leader from a country like Israel to his inauguration -- without inviting the leaders of all those countries who truly are our allies -- is political lunacy.