Well, actually, yes it does.
The 14th Amendment provides that “no state shall ... deny to any person within its jurisdiction the equal protection of the laws.”
Thus states are now as bound by the Bill of Rights, including the 1st, as is the federal government.
Islam is indeed a religion. It is a violent and political religion, but it is certainly a religion.
Islam had been around for over a thousand years when our Constitution was written, and you may note that the Founders did not write the 1st to exclude it.
Realistically, there would have been no reason for them to do so, as they would not have foreseen Muslim immigration.
But various states allowed religious freedom with various exceptions: Catholicism, non-Christian religions, etc.
The Founders ignored all that and went straight to “religions,” which meant ALL religions, even those that are violent and political and inconsistent with American values.
I have no objection to changing the Constitution via amendment to exclude Islam, but I object strongly to any government agency being given the power, by ignoring the plain wording of our present Constitution, to decide what is and is not a “legitimate religion.”
Actually - those prohibitions exist in every state constitution today also. Most states couldn’t come into the Union unless they adopted them. If a State Constitution prohibits it, then every agency of the state is bound by the same limitations, which includes Counties, and cities.