Based on the existing answers, and some further thought I’ve given things, this is the answer I’m currently favoring. Most of the examples I use are in spoilers, since they are just examples and the rest of the answer should be readable even if you haven’t gotten that far so long as you trust me about the examples, but a major point in “The Pointy End” (S01E08) is too large to spoiler.
Traditionally, kings could not release people from oaths.
The actual wording of the oath is very personal. It’s about what you will do, regardless of what the other does; in fact they’re often sworn to the gods rather than to any person—oaths are not contracts.
Moreover, we have examples of honorable characters ignoring attempts to be released from vows. Brienne of Tarth, for example, flat-out ignores attempts to release her from her vow
even though both Arya and Sansa each independently told her she was free to consider her vow fulfilled and to go away.
(“The Children” S04E10, “High Sparrow” S05E03)
And for that matter, Podrick Payne also remains quite stubbornly insistent to do his job even when unwanted,
Brienne just can’t get rid of him no matter how hard she tries,
(recurring point, first in “Oathkeeper” S04E04, until “High Sparrow” S05E03)
and that wasn’t even something he swore to do. And neither side of that debate feels that the subject of the duty is allowed to free Pod from it if he feels it remains his duty. Instead she repeatedly tries to convince him that any responsibility he had toward her was long since fulfilled.
Oaths are seen as sacred to both old gods and the new, and spiritual men throughout Westeros comment on that. The High Sparrow takes it to ridiculous extremes with his “star chamber,” but his entire position is based on tradition and faith, and no one even once questions his claims as to what tradition and faith had to say on the matter.
At least in the books, only the High Septon could release vows before the gods.
(Those vows sworn to the old gods, presumably, either could not be released at all, or required someone else of analogous authority.)
When Tywin Lannister suggests that Jaime step down from the Kingsguard,
since he lost his hand fleeing the Riverlands and King Joffrey Baratheon had just been murdered,
(Storm of Swords)
Jaime points out that Kingsguard vow to serve for life, and Tywin responds that “A suitable gift to the Faith will persuade the High Septon to release you from your vows.”
The only precedents available were recent, and viewed poorly.
The example of monarchs releasing characters from oaths all start with Ser Barristan Selmy: King Joffrey Baratheon released him from the Kingsguard, despite Barristan’s oath to serve for life. Barristan the Bold was the first to be “honorably discharged” from the Kingsguard in all history; all prior examples of someone being removed from the Kingsguard were cases of Kingsguard who broke their oaths and either deserted and fled, or were sent to the Wall. Removing Selmy was widely viewed as dishonorable, of questionable legitimacy, and as defiance of tradition. It did not sit comfortably for any of the honorable folks of Westeros.
The Baratheon and Lannister monarchies were then replete with oaths absolved by royal decree, e.g.:
Tywin Lannister offering to remove Jaime Lannister from the Kingsguard in King Tommen Baratheon’s name as part of a deal to spare Tyrion, King Stannis Baratheon offering to remove Jon Snow from the Night’s Watch so he could take Winterfell and become Warden of the North.
(“The Laws of Gods and Men” S04E06, “The House of Black and White” S05E02)
But both House Targaryen and House Stark were in a state of declared war against the remnants of that monarchy, and dishonorable behavior on the part of the Lannisters (including, but by no means limited to, playing fast and loose with oaths) was one of the primary causes of much of the conflict on Westeros for several years prior to “The Dragon and the Wolf” (S07E07).
And, even assuming the line from Storm of Swords that didn’t make it into the show nonetheless remained true, the High Septon was out since
there was no High Septon after Cersei had the Great Sept of Baelor blown up along with the High Septon and anyone who might have succeeded him; several characters comment on the lack of any High Septon.
(“The Winds of Winter” S06E10)
So for Daenerys Targaryen to release anyone sworn to her, even for their mutual political benefit, would not have been valid traditionally. The only precedent they would have had for such a move would be the Baratheon–Lannister monarchy they were opposed to, both in a literal, military sense and perhaps more importantly, in a moral sense. It would have been very much counter to Daenerys’s stated goal of “breaking the wheel.”