There's a blockage in the drain somewhere below (or in) your unit. A blocked vent won't cause that issue. If it didn't come up in the tub, it would just find the next place which might be the toilet or a sink and since they're not as large, it might overflow onto the floor. The drain cleaning job didn't go far enough, or wasn't done well.
If you were to run water long enough, your tub and likely other things (sinks, toilets, etc) would back up, too.