Some thoughts about the ideas thrown around here:
First, the player stations are already bought via the equipment screen. Then a missions window pops up (I used the template from your five thargoids aquatics equipment for that, T) that says something hopefully atmospheric about the bought station and shows it off.
The player stations are a piece of pseudo-equipment each (each one of the 192 stations is unique) that defines a mission variable when bought and is immediately removed.
The reason is as Screet said: Otherwise, the player stations wouldn't stay if you change ship, and they could be damaged in combat
As the mission variables now contain the information if and which stations you own, this info is independent of if the ship is changed/sold and it is saved in the savegame. Everything is then derived from there:
If the mission variable for a given station is on ("1" instead of undefined "null"), you are getting credits and legal consequences once each week, each station (its type, location system & galaxy plus income) is listed on the missions F5-F5 screen, the "sell" option for it/them starts to appear on the equipment screen and if you visit the system in the correct galaxy, it is spawned via planetinfo.plist there.
The mission-offering never figures into that, nor should it! Too much could go wrong there, Eric wrote about this, and it would be a pretty unwieldy way imho to list your owned stations like that.
As I said: I am happy to include any alternate listing solution that, at any time, shows the player all his stations, their type, average income, system & galaxy - as long as this is not done on the equipment screen or as a mission offering.
Hope to have consolidated the relevant information as food for thought
L