I've seen that for pretty much every faction I have ever been at war with. I believe it is just how the AI perceives everything. Could be a bug in the code that the AI can see whatever is in its explored portion of the galaxy. Or could just be a way for the AI to have an easier time during war.

I have seen this also.  The same thing happened in GalCiv2.  The AI is supposed to be good at predicting what you will do, but I think it's more than that.  I think that when Ship A decides to go after Ship B, Ship A will then know where Ship B is at all times.  It doesn't matter how far away Ship B is or if it is visible.  It then stays that way until something happens to break the connection, such as Ship A changing to a different target.

I have also seen it when I tell one of my ships to go after an enemy ship.  My ship will then follow the enemy ship and adjust its movement even if the enemy ship can no longer be seen.  It is especially noticeable if my ship is slower.  It gets further and further behind but still knows where the enemy ship is and will chase after it.

Draginol/Frogboy addressed a point about the AI in the Steam Forum for GCIII:

"...The AI has range on it same as you. No it can't see everything.
Once you play more, it'll be more obvious that the AI doesn't see everything. If the AI could see where planets were to colonize it would be devastating. But it is really good at guessing where things are going to be once they enter their FOW."

And this:

"...As the guy who writes the AI, I can tell you that the AI doesn't see everything. A huge bulk of the work goes into the AI trying to guess things. If the AI could see everything it would absolutely crush the human player because it could organize perfect fleets and never end up getting its ships ambushed. 
If you want to put a link to your saved game, I'm happy to look at it and find out either (a) how it sees the ship or ( fix whatever bug that is allowing it to see it.
The FOW system you have is the same FOW system they have (managing AI FOW is literally the most expensive CPU/memory task in the game). "

And this:

"Most of what the AI does is try to guess things based on information it has. There are a lot of things that are non-random that people think are random that the AI will take advantage of (such as where trade ships go, where in a star system the habitable planet would be, how many turns until a certain unit will go. Where a ship will be in N turns.

"A human player could, in theory, do the same thing but they typically don't. Few players will click on an opponent ship, make a note of how many turns they take and remember that in 8 turns it'll be at spot X because it was going in direction Y. But the AI can and will.

This is why I tend to be a little defensive on this issue because if the AI could just "see" everything, it would be so much easier. Nearly a gigabyte of memory is consumed on a large map because each AI has to keep track of its own FOW.

Now, if you increase the difficulty to a certain point, the "sensor range" of the AI grows and grows until it can "see" everything. But that is documented as part of the difficulty levels.

When I see someone saying how the AI is "dumb" because it sent a transport or a colony ship right through some enemy units I cringe because that same player will later say the AI can see everything. Obviously, both cases can't be true.

That isn't to say that it's impossible that there's some bug in the FOW system. But every time I've gotten a saved game from someone who claims the AI was cheating, the debugger will tell me which AI player tagged it and when they tagged it and started trying to track it.

What most people don't see is that the AI is almost always wrong (>80%) it's just that they see the AI ship moving in a strange way. But even when it's "right" it's usually not right. It guessed a region of space and then course corrected when it got into sensor range."

http://steamcommunity.com/app/226860/discussions/1/1489992080511310403/

luceo non uro
 

The AI may or may not have the same range that I have because they may have different abilities and  research different techs.  That is not the problem.  The problem is that seem to ignore the range limit and send ships across the galaxy.  This is most obvious on Huge and Gigantic galaxies if you start near the edge of the galaxy.  The AI on the opposite side of the galaxy will send freighters to your planets when your freighters can't even reach the center of the galaxy.

When GalCiv3 first came out there was a problem with AI on the other side of the galaxy declaring war on you then doing nothing.  This was because they didn't have the range to reach you and didn't know what to do about it.  This was solved by teaching the AI to build starbases to bridge the gap.  They were very good it.  In Crusade the AI on the other side of the galaxy will still declare war on me, but now they can reach me without building starbases.  I can't reach them unless I build starbases.

Once I had what I thought was enough evidence of the AI ignoring range limits, I reported it on Ticket BKL-489-10361 in December.

When it comes to the AI being dumb or not seeing everything, I believe they don't see everything, at least most of the time.  I have settled planets that were near an AI starting star system that the AI could have settled long before I got there.  I feel that means they explored in the other direction and missed the planets, just like I have done more times than I like.

The situation where an enemy AI ship can track one of my ships all over the galaxy seems to be different than just seeing everything.  Once the ship decides to attack my ship, it seems to know where my ship is at all times and can follow it anywhere.

That is what I see.  There is no way they should know where my ship is, but they do.  I can make multiple changes in direction or stop for a few turns, and they still come directly at my ship.  This is not the same as the AI seeing everything.  It is one AI ship following one of my ships everywhere.

I do need mention one time in the original GalCiv3 where it was "obvious" the AI was cheating and seeing too much.  I was attacking the Iconians, and they always seemed to know where my ships were coming from.  I eventually noticed the icon of a scout ship when I was zoomed out.  That shouldn't have been enough to let them see me, but I clicked on it to see what I could find out.  I was shocked when the type of scout was Iconian Spy Ship!  That was a ship I had designed several games earlier when playing as the Iconians, and it had a very large sensor range.  So, instead of cheating, the Iconians were using my own ship design against me!  I was somewhat embarrassed.   I didn't solve the problem by destroying the ship because they had at least one other Iconian Spy Ship.  We eventually made peace.

I mention the Iconian episode because the AI can be tricky at times as it uses what is available, so a Human needs to be very sure before accusing the AI of doing something it shouldn't be able to do.  I have been careful to make sure I am actually seeing what I think I am seeing, so I feel sure that the AI sometimes does some things it shouldn't be able to do.  This is what I have submitted tickets to report them.

I guess now is a good time to mention that the AI ships ignore movement penalties that some tiles have.  This includes asteroids, all types of nebula and tiles in the area of effect of starbases belonging to races that have the Vigilant ability.  This is also included in Ticket BKL-489-10361.  This did not occur before Crusade.  I think this and the unlimited range problem are like the money bug that appeared when Crusade was released.  The money bug was noticed quickly and fixed, while the movement and range bugs were noticed recently and have not been fixed.

When you select your fleet it should have some kind of sensors report that lets you know if you have been spotted/detected by enemy sensors. Some text like "Fleet Detected by Enemy Sensors" shown near fleet details. Would give more confidence navigating undetected past enemy fleets and also that AI is not fov cheating.