Does Fleet equipment benefit bases?
A quick test using the editor suggests that fleet components do in fact work on stations. The test consisted of 4 station duels, with each pair of stations consisting of nearly-identical stations where one station is a copy of the other but with Fleet Targeting and Fleet Countermeasures added on top of what was already there. The stations were armed with 20 Titan Beams and 5 Phasor Lances, and had 10 Meridian Shield Generators and 30 Ultra-Dense Armor plates, as well as 3 command centers, 3 hyperfusion reactors, 5 massive cargo bays, 5 docking bays, no standard targeting or countermeasures componets, no damage control or repair components, 2 ion defense components, and sufficient life support and hab components; there might be a few other components involved that I don't recall, though none would have contributed directly to the combat performance of the stations. The stations which were given the fleet components received 3 of each fleet component. The test results were that in all four duels, the stations with fleet components destroyed their opponents; in no case did any of the stations equipped with fleet components lose more than 25 armor plates. The four stations without fleet components were transferred to a pirate faction using the editor, after the pirate faction had been upgraded to tech level 7 by means of the editor; the other four stations remained in the player empire's control, with the player empire having started the game at tech level 7 due to game settings. All stations were spawned into existence in deep space by means of the editor, and had full fuel and shields at the start of the engagement, and the player empire had not completed any wonders or research since game start (the duels essentially took place at game start) and did not to my knowledge have control of any special ruins or other bonuses which may have affected results.
Be warned that this is a very small sample size and that the test was not particularly well controlled (station sizes were not identical, stations only had the minimum number of reactors to meet power requirements and so would begin to suffer performance degradation as soon as one reactor was destroyed, station designs used weapons which were not necessarily guaranteed to overcome the reactive rating of the armor used by the opposing stations, stations were not designed in such a way as to ensure that the duels would last a long time to reduce the impact of luck upon duel outcomes, no particular care was given to ensuring that each dueling pair was separated by a similar distance as the other dueling pairs though all dueling pairs were guaranteed to be out of range of the other dueling pairs and each member of any pair was within phaser lance range of the other member), nor was there any check for bias (no control group to see if player-empire stations are more likely to win than otherwise-identical pirate-empire stations, no test with pirate-empire stations with fleet components against player-empire stations without fleet components, no test group to check if relative position of the stations mattered, no test to see if order of station placement mattered).