The announcement by the defence ministry late on Thursday followed years of speculation that Kainerugaba, whose social media outbursts have sparked diplomatic uproars, was being groomed for the top job.
Although the 49-year-old general has in the past denied claims he intends to succeed his father -- one of Africa's longest-serving leaders -- he has enjoyed a rapid climb through Uganda's army ranks.
In a now-deleted post on X last year, Kainerugaba said he intended to run for president in the 2026 elections.
He also appeared to take a dig at his father, writing: "How many agree with me that our time has come? Enough of the old people ruling us. Dominating us. It's time for our generation to shine. Retweet and like."
Following a row in 2022 over a post by Kainerugaba threatening to invade Kenya, Museveni, 79, sought to rein in his wayward son by telling him to stay off social media when it comes to affairs of state.
Museveni, who apologised to Kenya over the outburst, has nevertheless defended his only son as a "very good general" and promoted him to that rank just days after the row erupted.
Thursday's announcement was part of a reshuffle within the government, with the former army chief General Wilson Mbasu Mbadi appointed to serve as the junior minister for trade.
Opposition politicians condemned the move to promote Kainerugaba.
David Lewis Rubongoya, secretary general of the National Unity Platform (NUP) party, called it "a sad day for our country".
Another opposition lawmaker, Ibrahim Ssemujju Nganda, accused Museveni of treating the country as "a personal enterprise of his family".
"Ugandans need to rise up collectively and oppose Museveni's family dynasty because we own this country and not Museveni alone," he told AFP.
- 'Baby despot' -
As a high-ranking military official, Kainerugaba is barred from speaking publicly about political matters but has frequently waded into such discussions, causing diplomatic headaches for Uganda.
His tweets in support of Tigrayan rebels in Ethiopia angered Addis Ababa, while his thoughts on Russia's invasion of Ukraine and the 2021 coup in Guinea also raised eyebrows.
Last year, he declared that Uganda would "send soldiers to defend Moscow if it's ever threatened by the Imperialists", chiding Western nations for sharing "useless pro-Ukraine propaganda".
He had also attacked the Ugandan government for failing to rein in corruption after an investigation accused cabinet ministers and other officials of stealing iron sheets intended for villagers in the poor northeastern region of Karamoja.
Thursday's reshuffle also stripped Agnes Nandutu and Mary Gorretti Kitutu, two cabinet ministers currently facing trial over the alleged theft, of their positions.
To many Ugandans, Kainerugaba's position as heir apparent has been obvious but the government has in the past taken a harsh line against anyone discussing the matter.
In 2013, police shut two independent newspapers and two radio stations for 10 days after they published a leaked confidential memo by a senior general alleging that Museveni was grooming Kainerugaba to succeed him.
Many of Museveni's former allies, including his personal doctor Kizza Besigye, have fallen out of favour with the president over Kainerugaba's elevation, which has also aroused the ire of opposition politicians and government critics.
Prominent author Kakwenza Rukirabashaija fled to Germany in 2022, alleging that he was tortured in custody on charges of insulting Museveni and Kainerugaba, who he has described as "obese", a "curmudgeon" and a "baby despot".
gm-amu/gil
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