Is x(x+6)=0 the same as x = 0 and x + 6 = 0?
In a way. To say that the single equation on the left is the "same" as the two other equations is not technically true, but it does have the same solutions as the others.
Actually, there is no value of x such that x=0 and x+6=0.
x(x+6) = 0 is true if x=0 OR if x+6=0.
That is why we always set everything to zero when we factor an expression. If the product of factors is zero, then that means that one or more of the factors must be zero.