In our group, we enjoy playing a good deal of demigod games with character Cults, ranging from Exalted (1, 2), through Godbound (1, 2), and to a smaller extent even Chronicles of Darkness could fall under this umbrella (in CRMB one of the PCs runs a masonic society which basically is a Cult to their supernatural self). While a demigod Cult is pretty much a staple of the genre and something that can be pretty interesting if done well, they often bring some issues whenever they appear.
Roles of cults in RPGs
As a baseline, Cults in RPGs generally come with some mechanical benefits to the player. In Exalted they give you some free Willpower, in Chronicles of Darkness they give you some free Merits, in the New Gods of Mankind they are key to getting faith to power your miracles, and in Godbound Cults give you more Dominion to change the world and come with some other benefits.
Beyond the mechanical, Cults are an expression of the demigod PC and their beliefs on how people should behave. Swallows of the South's Godwin was a musician, so his cults were his groupies. In Princes of the Universe, the Royal was all about being a merchant prince and trade, so his cult focused on heavy capitalism, paying taxes and accruing as much wealth as you could to get a high score when you die. In the same series the Majestic was a narcissistic scholar and his cult focused on people becoming educated. He was also heavily themed after mesoamerican themes of Exalted, which meant a lot of human sacrifices. But since only the best would do for him, the sacrifices had to be of the best examplars of his cult and willing because Majestic was against slavery.
Similarly, Cults are a way for the players to shape their world - to introduce a new religion into the world and possibly affect a large number of NPCs. Similarly, the Cults are often a resource the PC can tap into to engage the world more broadly - if you have a kingdom that worships you, you can rally troops from it and send that army to conquer other territories for you after all.
There are some problems that appear when you start dealing with the Cults in practice however...
Possession of NPCs
In demigod games we ran we often ran into this situation - a new mortal NPC is introduced, they join our society, and then comes the question of who will "own them" (so to say) by which Cult they will join. After that the PCs can get possessive over the NPCs that are their worshippers, and those NPCs rarely interact with PCs that are not their chosen deity.
For example, in Princes of the Universe a young girl named Conna is one of the last survivors from Wanderer's village. Early in the series the Majestic takes her under his wing to educate her properly and so on, later making her an important figure in his cult. Pretty much from that point she doesn't interact with anyone else anymore, and later Majestic basically refers to her as his adopted child. Similar patterns also appear in The Living Years where various mummified saints are raised from the dead and become worshippers of the Litch King and not interact much with other PCs, despite one of them being a devout of the faith they were the saints of, etc.
It's perhaps a bit subtle at times, since in a lot of situations you will have certain PCs and NPCs gravitate closer together and you may have NPCs that don't interact with anyone but one PC in general, but an NPC becoming someone's worshipper is certainly a strong indication to other players to not mess with them because they "belong" to that player.
On a larger scale, the same principle applies whenever the players visit a new location. Since some systems like Godbound punish you for divvying up a population between characters (Cult scale is exponential, but breakdown is linear. A Scale 1 Cult is 1000 people, a Scale 2 town is 100k people, but you can either get one Scale 2 Cult out of it, or two Scale 1 Cults, and combining two Scale 1 Cults from two places doesn't give you a Scale 2 Cult), meaning you often have to pick who owns what new town you visit or save. It starts to feel very transactional, especially when players start expecting towns to convert to their religion after saving them.
On a similar note...
Transactional worship
Religion defining the NPCs
A character can appear in the story and later join one of the PC Cults, or they can be introduced as someone from the Cult outright. In the second case especially (although not exclusively) the NPC's involvement in the Cult tends to define their personality. Godwin's groupies exist only as his fans with a little bit of individual mannerisms sprinkled on top. In CRMB every member of the masonic Cult tends to act in a similar, scheming way. etc.
Sure, sometimes when you create disposable NPCs, they tend to be one note, but unfortunately it seems with NPCs that are a part of a PC Cult that note tends to gravitate on what religion they subscribe to. This can make them a bit less interesting than if one would build characters first and then figure out if they'd follow any of the Cults.
In a similar vein, it is also rather easy to portray these Cults as a homogeneous group of people, rather than a collection of individuals. They tend to be characterised as a collective, display uniform traits and generally just be that one-note character from the previous paragraph smeared across a larger group.
Of course, this can come down to how much time and effort one wants to devote to fleshing out the characters. There is nothing stopping you from adding more depth and nuance to Cults and its members, but it takes that little bit extra effort over going with the flow of the least resistance and using stereotypes when describing the NPCs.
Mechanics influence characters
As discussed before, the mechanics of a game inform the playstyle of the characters, and the same can be said for Cults. I've noticed this especially with Godbound - in this system the characters can either be a demigod with a cult, or a free divinity that doesn't have worshippers to suit your playstyle. However, mechanically, having a cult more often than not is a better choice - you get more Dominion out of it, you get a neat Faction you can use to do your large-scale bidding, and generally have more stuff to interact with. Mechanically, the game is rewarding you for having a Cult, which means almost every character in the game will end up having a Cult (in our multiple campaigns using the system, totalling to about 15 characters, only one PC didn't have a cult - Adina from The Living Years, and that was only because I ended up insisting she wouldn't get one if it couldn't be a part of the religion she followed herself). This in turn changes the sort of characters you play - you won't want to play someone who is selfless and doesn't accept people worshipping them because that would put you at an advantage, so you make a character that would want the worship.
One solution - make it all cosmetic
One possible solution to a number of the listed problems would be to make the Cults a cosmetic thing to a character, rather than something that has mechanical benefits. A narcissist demigod that demands adulation can still make a Cult because that suits the character, while someone that doesn't feel the need won't feel bad for missing out on the mechanical benefits. Now because Cults stop being such a dominant thing every player has to focus on, the NPCs are under less pressure to pick a side and they can remain their own people. Being praised as "the town's hero" for saving it feels a bit less of a commitment than having the town worship you as their deity of choice.
Conclusions
Character Cults can be a really interesting part of a demigod game. They can serve as an extension or a compliment of any character, a way for them to express themselves on a large scale. However, it is very easy to fall into traps of defining NPCs by the religion they follow, and to get territorial and transactional about individual and group NPCs and which bucket they will fall into. When the system promotes having Cults, you also tend to see characters more skewed in that direction, changing what kind of PCs you see in those games.
Ideally, you would focus on making NPCs defined people first and then putting them in a Cult if they fit, and you would have a system where choosing to have or not to have a Cult would be balanced to encourage making characters that fit what the players want to play, rather than rewarding one kind of characters.