In a proper trilogy the middle story both raises the stakes and creates an indirection that is paid off before the final installment.
At the end of Identity Bourne has two problems, he's got a powerful enemy in Carlos and the (justified) distrust for the CIA and the American government for using him as a pawn in their game to bait out Carlos. A lot of the tension of the first book is also tied up in his amnesia, especially as he's the only one to recognise Carlos's face and doesn't remember everything of his past with the CIA.
Supremacy resolves some of the tension of his Amnesia, opening the door for him remembering Carlos later, with the help of Marie. This book also works on ramping up the tension between Bourne and the CIA (the misdirection) as they continue to manipulate him - making it look to Bourne like they're the biggest threat to him and his family. However as Bourne works with Conklin, a figure from the previous books he partly resolves the CIA side of the tension from his past, whilst leaving the consequences from his actions as Jason Bourne unaddressed.
The actual plot in the second book seems to not be relevant, nor is how Bourne uses his hyper-competency to resolve issues. It's the pivot from running from the CIA to trusting some of them, as well as the character development for all involved. It's what switches Bourne from facing one issue (CIA, Memory, etc) to another - his past - hinted at by the 'imposter Bourne' using the real Bourne's past deeds to his advantage. We also see McAllister, symbolising the American government, is shot but recovers implying that this threat is not the one that Bourne should focus on.
Ultimatum then calls back to the first story, while the plot directly tackles Carlos, Bourne is the one who reaches out to Conklin and the CIA for protection. Part ways through this resolves the ramped tension from the second book, i.e. in the second novel it looks like the CIA pose the biggest threat (the second books misdirection), whereas they now turn out to be allies. Only after that misdirection and tension is resolved, can the originating tension (his past, symbolised by Carlos) can be addressed.
Interestingly, as an aside, it looks like Ludlum came up with the first two books together, only later adding on the third installment:
Robert Ludlum gave two interviews to Don Swaim of CBS: in 1984 and then two years later in 1986. Ludlum discusses how he came up with the first two novels in the Jason Bourne trilogy – The Bourne Identity and The Bourne Supremacy.
To provide a comparison, and help explain, consider the Harry Potter series (purely because it's a well known series, no other reason). Let's split it up into three parts as though it was a trilogy and not seven books:
- Philosopher's Stone, Chamber of Secrets: These serve to show Voldemort as, although alone and desperate, the primary antagonist. The plot in each is relevant here as they both set up the character of Voldemort.
- Prisoner of Azkaban, Goblet of Fire: Book three gives us the first misdirection of the series (we're ignoring Snape's accusal in book one, as that serves more towards the book's three-part structure, but doesn't really feed into Snape's part later on) where Sirius looks to be the primary antagonist whereas it's the Magical Government run by Cornelius Fudge that proves to be the real opposition, and importantly continue to be so later on in the series. In Goblet of Fire, the Triwizard Tournament plot itself is a convenient vehicle for the overarching series but isn't itself significant until the last moment. And in fact that last moment, with the port key, could have happened at almost any point. This however shows Voldemort personally raising the stakes for the first time. He's no longer alone, and he's actively taking steps to defeat Harry Potter. All the while the magical government are making things increasingly difficult for the trio. Neither plot is significant (except the very, very end of Goblet of Fire which could have been written at the start of Phoenix without involving Harry) to the overarching series, but other factors play their part like character development and motivation.
- Phoenix, Half-blood and Deathly Hallows: From the very beginning the trio and the Order are working outside of the government (sidestepping the misdirection) and solely focusing on Voldemort. The quest to collect the horcruxes (in response to Voldemort raising the stakes) has started and the series is drawing to a close.
Again: The plot of the middle book(s) is not relevant, but it's used to misdirect, raise the stakes and switch focus.