My codo was built in the 40s and I'm having a problem with the second floor plumbing . The vanity is backing up into the bathtub both on the second floor. The toilet seem to be fine when flushed.

You have a blockage downstream from your tub. The solution would be to snake the tub drain line. From the overflow. (see image).Let me show you how,
Unscrew and remove the overflow plate,(The one with the drain lever) and pull out the tub stopper. You now have a clear shot at the trap. The way you were going in through the drain it ran you square into a tee that you can't get around. Now feed the cable( 1/4" best or 3/3" with a small tip. You can rent a Ridgid K50 sink cable.) About 2' down you will run into the bend of the trap. Crank and push at the same time to work around it. Your blockage will be found in the lateral pipe going to the stack. You shouldn't have to put out more then 10 or 12 feet. I have found tub and shower clogs to be mostly hair and the worm should auger into it and pull it back. After you snake pour a few pans of boiling water down the drain. This will loosen the grease and flush the mess out into the main. Hope this helps and thank you for rating my reply. TOM

Because you have clogged common drain. Drain fills up with waste water and starts rising until it reaches first open outlet - which is tub drain. They you start seeing water backing up. Sink has drain 17" high, therefore you will never see it backing up in the sink. Hope that explains.

Do as Tom suggested. Back to you. Milo