I had never before heard of this Jakob, so thanks for bringing him to my attention!
Some brief research, however, reveal that most Danish historians of the period do not believe him to be legitimate. Jørgen Nybo Rasmussen, who made the claim in 1986, uses some questionable sources and flawed methodology in his argumentation. Jens E. Olesen (a very competent and skilled historian of medieval Denmark) finds the conclusion to be highly problematic. Queen Christina's household documents make no reference to a second living son, just as no other contemporary sources do. It would be very weird for a legitimate son to go under the radar like that, especially considering that such princes were valuable commodities in pretty much all aspects of state (which to me is probably the most obvious argument against his legitimacy).
However, if we accept the premise of Jakob indeed being a legitimate son of Hans I, quite a few variables in early 16th-century Scandinavia are definitely altered. Therefore, I don't expect events to follow the same trajectory as in OTL. Of course, a lot depend on the exact relationship between Christian and his younger brother and what career, his father groomed him for. Although Hans tried to pressure Frederick I into joining the Church OTL, that was more an example of the animosity between the two brothers (in the run up to and following the division of the Duchies. Especially after Frederick tried to claim fiefs within Denmark at the 1494 parliament, relations soured between the two). As such, I don't think Jakob would naturally be destined for a career within the Church. He would, more probably, be invested as Christian II's statholder in Norway or the Royal Duchies. Christian II was a very volatile character in OTL and he might have alienated Jakob during the course of his reign (especially if we stick with the Dyveke debacle and the failed invasions of Sweden). On the other hand, he might just as well end up a die-hard partisan for his brother (like Henrik Gøye or Søren Norby).
Jakob is only three years younger than Christian - which makes him a possible candidate for the throne when Hans dies in 1513. We know that there already existed some kind of movement plotting Frederick's election at this time, but at that point Christian's uncle was simply the only other option. With Jakob alive, the conspirators would be in somewhat of a conundrum - their whole argument was that Frederick was "... born of true Danish blood" - i.e. related to the royal family. Jakob is too, and he's closer to the line of succession to booth. However, I don't think 1513 is the year the councilarist opposition would make a move, although the presence of an alternative heir would possibly put extra pressure on Christian II when it comes to negotiating his accession charter.
If we fast forward to 1523, there are also a few things needed to be taken into consideration. The royal part of the duchies would most likely be divided between Christian and Jakob, which diminishes the royal authority in Schleswig and Holstein, but leaves Frederick's Ducal domains intact. What made Frederick attractive as a pretender was not just that he had a claim to the throne, but also the fact that he had a quite strong powerbase in the Duchies. Jakob might have a better claim on paper, but he does not have the same access to resources as Frederick does. As I said before, a lot also depends on Jakob's character. Is he a "progressive" like his older brother or a "conservative" like uncle Frederick?
