I'm sure there will be good answers for how to houserule this spell and bring it under control, so I'm going to go on an extreme tangent and suggest something else entirely:
The cleric clearly wants to be a rockin', evil-burninating machine. Awesome! Roll with that. Bring out more, nastier evil outsiders and help make that cleric player feel awesome. As the DM, sometimes countering the spell's effectiveness is going to be what delivers the most fun. Sometimes throwing hapless demons at that Celestial Brilliance just to watch them burn like moths in a candle is going to be most fun.
No, darkness doesn't cast a shadow.
At least, it doesn't without a particular DM making it a table rule — which would be a totally reasonable ruling to make. To tackle the question though, we must consider the default baseline from which such rulings would be made, avoiding assuming rulings that give us a result from circular reasoning.
Darkness not casting a shadow is non-intuitive, but magical darkness is inherently non-intuitive and operates by its own idiosyncratic rules.
I've emphasised the relevant parts of its effect description (PBR p. 86; PHB p. 230):
Magical darkness spreads from a point you choose
within range to fill a 15-foot-radius sphere for the
duration. The darkness spreads around corners.
A creature with darkvision can’t see through this
darkness, and nonmagical light can’t illuminate it.
The stipulation that it's effect is limited to a radius of 15 feet, and that its effect is to prevent illumination, not penetration of light, means that it does not and cannot alter light levels outside its area of effect.
Contrast this effect description of darkness with that of fog cloud: the latter's description simply says that it creates fog in a given area, and then leaves it as an exercise for the reader to extrapolate from our shared understanding of real-world fog — such as the fact that fog impedes the passage of light — with the help of the game's specific rules for obscured vision. Darkness, by contrast, does not try to leverage our intuitions about "magical darkness" — as none can be expected of us1 — and instead gives specific effects. These effects are sufficient for most groups, so that groups who don't care (or who prefer a non-scientific fantasy experience) can get on with the game without having to tackle the question "but how does it work?!" with lengthy deliberations first.2
It could be argued that the line "A creature with darkvision can’t see through this darkness" means that light is entirely prevented from passing through. However, such an extrapolation would be a ruling limited to the ruling DM's game, as other DMs will not all also make the same ruling. (Since how darkvision operates is idiosyncratic and non-lawlike in the same way as how magical darkness does, we cannot reason from how it operates to how darkness operates, without a DM adding even more interpretive rulings.) Since such a ruling is an addition and not universal to all possible DMs, and because magical darkness is inherently illogical, that extrapolation can't be considered the default, baseline case.
1. Those of us who have a specific metaphysical reality in mind that does create intuitions about how magical darkness operates are precisely those who would want to make rulings about darkness in order to satisfy our intuitions and to make the spell better match our imagined setting. This is exactly what the permission to houserule, that the game gives DMs, is designed to accommodate. For the rest of us, the game doesn't assume anything, and gives us bare effects for things that are not transparent to the usual intuitions of modern humans, like darkvision and the darkness spell.
2. In my own games, I'm likely to rule that it does cast a shadow because I tend toward classicism in my campaign metaphysics — if the issue ever comes up, which I expect it likely won't. Except that, if I'm running a romanticist campaign based on my ontological Chaos premise, I'm likely to say that it doesn't cast a shadow and neither does its contents, for no reason other than "because Chaos." So, there you go: it very much depends on the DM and campaign.
Best Answer
The first and most important thing: negating and countering/dispelling are separate. They don’t both happen from the same casting of the spell.
Negating
Negating is what happens when you cast celestial brilliance normally—it creates light in the area, except for the parts of the area that overlap with magical darkness. In those spots, instead of creating light, it negates the magical darkness (just as the magical darkness negates its light).
Applies to everyone equally
Celestial brilliance does what it does equally to everyone. There is nothing in the spell that allows the caster to limit its effect on allies. Anything it does to enemies, it does to allies too (and to the cleric themselves!). Likewise, anything it doesn’t do to allies, it also doesn’t do to enemies.
That includes the negating
So yes, celestial brilliance negates magical darkness, both from allies and enemies. But this is only “darkness,” not all dark or shadow-based magic. You are looking for the “[Darkness]” descriptor in the spell description, after the school. For example, deeper darkness has “Evocation [Darkness]” on the second line—that is what celestial brilliance interferes with.
If something has “[Darkness]” then celestial brilliance negates it in the area where they overlap. If something has something to do with darkness or shadow or whatever, but doesn’t have “[Darkness],” then it works fine in celestial brilliance.
Shadow spells and darkness spells are separate
“Shadow spells” are those that have “Illusion (Shadow).”
“Darkness spells” are those that have “[Darkness].”
The two aren’t the same! Shadow evocation is a shadow spell, but not a darkness spell. Deeper darkness is a darkness spell, but not a shadow spell. And celestial brilliance only negates darkness spells, not shadow spells.
So if the cleric’s wizard ally casts deeper darkness, the cleric’s celestial brilliance negates it where they overlap. The same for a shadowcaster’s black candle mystery used to create darkness (Tome of Magic pg. 142). And if an enemy babau demon casts darkness, celestial brilliance negates that too.
However, if the wizard casts shadow evocation, celestial radiance doesn’t do anything to it. And likewise for enemy wizards, they can also use shadow evocation within celestial brilliance with zero problems.
Finally, despite the name, shadow magic isn’t always technically in the “(Shadow)” subschool. Celestial brilliance doesn’t do anything to shadow spells in the first place, but if something does, it only works on the mysteries that actually have the “Illusion (Shadow)” school, like ephemeral image (Tome of Magic pg. 145).
And shadow mysteries like bolster—which is just “Transmutation,” no “(Shadow)” or “[Darkness]”—are not affected by anything that interferes with shadow spells or darkness spells, because it isn’t a shadow spell or darkness spell—it’s just a Transmutation.
Countering and dispelling
Countering or dispelling are alternate uses of the spell—instead of casting celestial brilliance, you can use the prepared celestial brilliance spell to counter or negate a [Darkness] spell of lower level. If you do this, you don’t get the usual effect of celestial brilliance—no light—and instead just eliminate the [Darkness] spell of whatever you countered or dispelled. Countering and dispelling are targeted effects, so you can (and probably would) choose to use them only on enemies.