No, Shakespeare alludes to multiple religions throughout all of his works. He also gives quotes to show that Religion has not aided him in anyway. For example, when King Claudius from Hamlet bursts into prayer in Hamlet (III. iii. 36)
Writers often try not to allude to the fact that they have a specific religion because it may persuade readers not to take interests in their writing solely because of their belief system. Shakespeare lived in an era where having a religion was almost a guarantee, so it is highly unlikely that he does not categorize himself under a specific religion. He merely hides his personal religious beliefs to expand his audience.
Yes but if Shakespeare were to want to captivate more people who, in this time, would almost always be apart of the same religion then why would he need to hide it? In a lot of other writing from this time period the author talks mainly of one religion and that is Protestantism. Why would Shakespeare not write about Protestantism more if that were the most dominant religion of his time? This give me more reason to believe that he did not belong to a specific religion.