Why was the movable type of the printing press such a breakthrough for publishing?

Movable type is not part of a printing press. Before movable type and printing presses, things had to be manually transcribed (copied by hand), which was extremely expensive. The press mechanized copying, lowering it's cost and driving down the cost of reproduction.

Arguably, there wasn't any publishing before movable type and printing presses.

It made it possible to produce identical copies faster and cheaper