Short answer: In history, Jiang Wei was well regarded by Zhuge Liang and ended up in his councils but not to any particularly special degree, no more than any other of Zhuge Liang's trusted or valued officers. The idea of being Zhuge Liang's protege is from the novel romance of the three kingdoms which had a major impact in how such figures became viewed.

History in part 1, novel in part 2

History

In 228, Wei was caught completely off guard by Zhuge Liang's first campaign against them, they had Xiahou Mao at Changan who was a friend and relative of Emperor Cao Rui but noted for not being militarily capable while experienced generals like Cao Zhen had been pulled to other fronts. Defences had been neglected as, with Liu Bei dead, Wei didn't believe Shu had anyone with the drive to attack.

Some of the frontier regions revolted, Tianshui, Nanan and Anding revolted to join the attacking army, Ma Zun the head of Tianshui was on a tour of inspection in the area with local officers like Jiang Wei with him. Ma Zun fled when it became clear the locals were rebelling, Jiang Wei surrendered to Shu either on local wishes or because he had been shut out by local towns.

An inexperienced Zhuge Liang lost the campaign with both the armies he sent out against Wei forces losing while Zhuge Liang was too far away to reinforce. The choice of his friend and protege Ma Su leading key army of Jieting instead of more experienced figures like Wei Yan, Ma Su then disobeyed orders on where to camp, got surrounded and supplies cut off by Zhang He before taking heavy losses. Generals were demoted or executed and Zhuge Liang, with not all the court behind him, demoted himself.

In such circumstances, the defection of a local note able was to be celebrated, ranks given to encourage further defections and attempts to highlight a rare boon of a disastrous campaign. However, while giving a marquis would be part of that, Zhuge Liang was impressed after meeting with Jiang Wei, writing to Zhang Yi (who was managing affairs at capital Chengdu) and Zhang Yi's assistant Jiang Wan

"Jiang Boyue (Jiang Wei's style) is loyally assiduous towards the affairs of the day and his thoughts are fine and exact. As for his qualities, Yongnan (style of Li Shao who had died at Yiling), and Jichang (Su's famed elder brother Ma Liang who also died at Yiling) are not his equals. He is a superior gentleman of Liangzhou."

On another occasion, he wrote "Wait till I have him train the five or six thousand men of the Central Hubu detachment. Jiang Boyue is very competent in military affairs; not only is he courageous and proficient in warfare but his heart is loyal to the House of Han and his talents combine those of other people. I shall instruct him thoroughly in warfare, and shall also send him to the palace to be received in audience with the Sovereign.'

While, due to needing to highlight what positives he could from that campaign there may have an element of selling this recruit as hard as possible, Zhuge Liang's plans for Jiang Wei were true. Jiang Wei got that audience for which he was promoted further, then Jiang Wei trained troops at the capital. The teachings aren't mentioned again and Jiang Wei was never linked as a student of Zhuge Liang (more seen as a student of the Zheng Xuan tradition) bar that letter.

We don't get much of Jiang Wei's till Zhuge Liang's final fatal campaign. Jiang Wei gets promoted twice, one would assume this was for good work and not being drawn out of a hat but Shu's records are poor. There was a records department but no history project to the equivalent of Wei and Wu with scholars like Qiao Zhou attached to education, there are a lot of gaps. Liu Shan's first Empress has a whole one paragraph, the famed general Huang Zhong biography is rather small

In 234, Zhuge Liang launched a joint campaign with Wu and Jiang Wei was part of his circle during it. With the Wei commander Sima Yi sitting back his defences at Wuzhang and Shu seeking a fight, Zhuge Liang sent a woman's dress among efforts to provoke a fight, the Emperor Cao Rui sent long-serving adviser Xin Pi with an order to hold the position with Jiang Wei remarking Xin Pi's arrival means they would not come out to fight

Zhuge Liang became mortally ill during the campaign and Jiang Wei was part of his circle to be given instructions as to how to carry out the retreat after Zhuge Liang's death. Jiang Wei was to be the second rearguard (or main one if Wei Yan elected to do his own thing) and when Sima Yi pursued on hearing Zhuge Liang was dead, Jiang Wei got the commander of the retreating army Yang Yi to turn about and draw up the army as if to fight which caused Sima Yi to halt. While some figures in the Shu army like Yang Yi and Wei Yan hoped to take command, Zhuge Liang had written to Liu Shan before (and confirmed in his comments to envoy Li Fu at Wuzhang) "If I your servant suffer misfortune, later affairs should be entrusted to Wǎn"

Jiang Wei had become a member of Zhuge Liang's council, someone who was part of the inner circle and trusted to act as the rearguard. We get his promotions and his role in final campaign but bar those two letters from Zhuge Liang from early on (and those were more about this new talent he had found), we don't get any sense of their relations on a personal level. Jiang Wei was one of Zhuge Liang's men but Zhuge Liang had brought through and promoted a lot of figures like protecting Jiang Wan from Liu Bei's anger, there were plenty of senior figures (like Yang Yi and Wei Yan) in the army who Zhuge Liang also valued. In that regard, Jiang Wei was not an exceptional figure in that nor was there any expectation he would take over, his relationship with Zhuge Liang was not held as something really worth highlighting.

Jiang Wei was well established among the army and at court, he would get an army under Jiang Wan to carry out one plan and under Fei Yi, would share the work of the Secretariat and became Fei Yi's (deliberately restricted) commander with some raids on Wei. Only when the defector Guo Xun assassinated Fei Yi in 253 to end the era of the four heroes (Zhuge Liang, Jiang Wan, Fei Yi and Dong Yun) did Jiang Wei become the military leader of Shu and the chief figure at court with the civil offical Chen Zhi beneath him and an understanding (along with the eunuch Huang Hao) formed between the two.

Michael Farmer in his work The Talent of Shu: Qiao Zhou and the Intellectual World of Early Medieval Sichuan does comment about how Jiang Wei may have been shaped by Zhuge Liang years later. Jiang Wei's desire for reputation and fame was considered one of the driving forces of his actions when he became Shu's chief general and led to some criticism from ancient historians. Farmer suggests this drive was because Jiang Wei was haunted by the popularity and reputation of the worshipped Zhuge Liang and wished to build his own reputation, to get out from under Zhuge Liang's shadow