To answer the OP's question directly (and trying hard to ignore the sub-infinite exaggeration in the OP's post):
It doesn't tell you exactly where you earned the fine, because fines apply to factions, not star systems. A faction might be present in dozens of star systems, and you can pay that fine in any one of those. Or they might be present in just one.
It's true, the game could be more useful in helping you locate specific factions. Having a "faction map" on the galaxy map would be darn helpful, not only in finding fine-sources, but in monitoring the spread of influence of "your" faction, or "enemy" factions, for those who care about the BGS. But we don't have that, so there's no point crying over eggs that haven't hatched yet.
To find factions, it's currently easiest and most helpful if you use external sites. I find INARA.cz most useful for locating factions - you can simply search for the faction and it gives you a list of all the systems they are present in, and where they have a controlling influence over a system or station.
For the precise four examples given in the OP's screenshot:
- The Pilots Federation Administration owns "Newbie space", and nowhere else. If your Newbie Space permit has been revoked (which I assume it has by now), you can't go there to pay those fines. Your only option to get rid of those fines is the Interstellar Factor.
- Ehecatl Free is, obviously, found in the Ehecatl system where it controls most of the stations. It is also found in Fjorgyn, Hollatja and LTT 8456.
- The Pilots Federation Local Branch owns the "protected" system of Shinrarta Dezhra. They also own numerous stations in several systems throughout "old newbie space", including Asellus Pimus, LHS 3447 and Eravate. They also own numerous rescue ships in Thargoid-attacked systems.
- AY Indi Limited is, naturally, based in the AY Indi system where they are currently the controlling faction. They are not present in any other star system.
The PFLB are also present in AY Indi, so you can "kill two birds with one stone" by going there and paying off both your PFLB fine and your AYIL fine.
