Bash regex =~ operator

What is the operator =~ called?

I’m not sure it has a name. The bash documentation just calls it the =~ operator.

Is it only used to compare the right side against the left side?

The right side is considered an extended regular expression. If the left side matches, the operator returns 0, and 1 otherwise.

Why are double square brackets required when running a test?

Because =~ is an operator of the [[ expression ]] compound command.