Do saltwater swimming pools produce the same asthma symptoms as chlorine pools do?

No, but you will get dehydrated, making you breath heavy.

I don't agree that you get dehydrated. The body fluids contain salt so in a plain water pool the body is taking on water because it contains salt.

The fluids prescribed for an isotonic solution to rinse the nose to keep the sinuses from either gaining or losing water would contain about 500 Gallons of salt/25,000 Gallons of water.

A salt pool uses less than 100 gallons of salt/25,000 gallons of water. So the body still absorbs water but less than with a plain water pool. So a plain water pool is more likely to cause the sinuses to swell than a salt pool.