After looking at it again spaceports and the specialized facilities don't stack with each other nor with themselves. If you need all three on the same colony and have workers available then the spaceport is cheaper at 3k BP vs 3600 BP for a full set of specialized installations. If you don't need all three then the specialized installations are always cheaper.
Shuttle stations however stack with freighter shuttles. That is, having a shuttle station counts as having one extra shuttle on every ship in orbit, including ships that don't have their own shuttles. Additional shuttle stations don't stack, however.
Are you guys chatting about the spaceport you can build in industry tab?
Yes were trying to determine if having more then a single port increases anything as there is some confusion about it.
I really don't see a reason to ever build shuttle stations anywhere since you technically need a cargo ship with shuttles to carry them to somewhere and at that point just use cargo ships with shuttle bays. I guess you could use them in a game similar to mine where I am not using mass drivers if you needed a fast cargo ship or something and wanted to save weight on shuttle bays..
It depends on how much stuff you need to move. A freighter with one shuttle bay will load or unload twice as quickly if there is a spaceport or shuttle station present, and the load time with two shuttle bays is reduced by 1/3. So if your freighters have one shuttle bay each and you are moving more than 500k tons* to a planet then you are better off moving a shuttle station there first. If your freighters have two shuttle bays then the tipping point is 750k tons.
*500k tons is 200 infrastructure, or 100 infrastructure and 1/5 million colonists. This makes a big difference for civilian colonization efforts and trade.