He didn't. He belonged to a company called The Lord Chamberlain's Men. This is like asking why Mr. Eastman called himself Kodak. He didn't. He called himself Mr. Eastman. Likewise William Shakespeare called himself William Shakespeare.
The Lord Chamberlain's Men was a company formed by Richard and Cuthbert Burbage and a number of other actors including Will Kempe and William Shakespeare. They were called that because their aristocratic patron (without whom the company would be illegal) was the Lord Chamberlain.
The Kings men
William Shakespeare first worked in Lord Chamberlain Men later called King Men
The acting company to which Shakespeare belonged for most of his working life was called The Lord Chamberlain's Men from 1594-1603, and The King's Men after that. Shakespeare was not the troupe's leader, but rather a partner.
The Kings Men.
Shakespeare was a member of the King's Men. They were an acting company and he was one of the actors. He was actually a charter member back when they were called the Lord Chamberlain's Men.
There was no acting company called the King's Men until 1603, and when it was started Shakespeare was one of the charter members.
Lord Chamberlain's Men is what it was first called.
Lord Chamberlains Men
Shakespeare went into business with his partners in the theatrical company The Lord Chamberlain's Men in 1594. Before that time there is no indication that Shakespeare was in business for himself as opposed to being someone's employee.
In Shakespeare's day, theatres were called playhouses. In fact, the name of the first venue for the Lord Chamberlain's Men was The Theatre Playhouse.
It was called the Lord Chamberlain's Men to 1603, and thereafter The King's Men until Shakespeare retired in 1613. I don't know where you get 1608 from.
Shakespeare spent most of his career with the theatre group called The Lord Chamberlain's Men and later The King's Men, where he worked as an actor and playwright.