If fleet admiral is the highest rank that a Starfleet officer can reach, which is seems to be, can a fleet admiral promote someone else to their own rank? Otherwise, who appoints/promotes people to these highest-ranking positions? The President of the Federation, who they report to, but who is a civilian? Or are they selected by their peers somehow?
At the highest level, promotion typically falls under the purview of the ultimate real authority to which a military reports (So putting aside situations like Canada or other constitutional monarchies where theoretically the monarch issues commissions and appoints senior military officers to positions, but really it's the elected government).1
In the DS9 episode "Homefront" President Jaresh-Inyo makes it quite clear that he has the authority to force Vice Admiral Leyton (stated in the background to be Chief of Starfleet Operations, but never made explicit on screen) to resign, and is stated to be Commander-In-Chief of Starfleet explaining how he has that authority. So, presumably, they that can dismiss is also they who can appoint in the first place.
In the Federation, we really have no canon evidence of how the civilian political system works, let alone the details of its control over Starfleet and when the civilian authority exerts practical control over positions and appointments and when it's considered an internal Starfleet matter that the civilian oversight essentially rubberstamps. We know there's a President and cabinet and a Council, but other than that it gets vague. Presumably there's some kind of Minister or Secretary that Starfleet falls under, but we're not given any information on whether the cabinet consists of Presidential appointees (like the US) or of elected Council members (like parliamentary democracies). Or perhaps there's a special group of Council members specifically responsible for Starfleet. We don't even know if there's a Federation wide election or whether, like in many countries, the President is elected from among members of the Council by the Council itself, acting as representatives of their planets and colonies.
We know there's certainly civilian oversight based on the trial of Kirk and Co in Star Trek IV, where the tribunal consisted of the Federation President and members of the Council. Compare this to say, the trial of Michael Burnham which was clearly an old fashioned Court Martial with superior officers being the ones running the show. The obvious difference is that Kirk was an admiral who, basically, mutinied as opposed to Burnham who was a commander, and having admirals running around not following orders is more of a problem than a commander.
1 Note that this is true of dictatorships as well. While there can obviously be exceptions--usually in smaller countries--Hitler, as an example, typically didn't care about who the Wehrmacht promoted to a battalion command, or who generals brought on their staff as assistants. Once you started getting in command of divisions and armies, and other senior leadership positions, then he weighed in. Stalin was the same.