Answering this on the clarification in comments by OP that:
the setting is an alternate reality and more fantasy than sci-fi; "Earth-like except as noted". The animals in question are "mostly" identical to real animals except for specific, limited differences; sapient, bipedal... and sometimes blue(ish gray). Less Star Wars, more Zootopia.
Other people have already pointed out that real world birds, plants, etc produce blue pigments.
I'll add an alternative Earth spin to this: way back in primordial history, the ancestors of the vertebrates didn't use hemoglobin (iron based) as the respiratory pigment in their blood, they used Hemocyanin, which is copper-based.
Hemocyanin is not as good as hemoglobin for transporting oxygen around the body, so you cover that with the fantasy aspect of your setting, and hand-wave it away.
Hemocyanin is blue when oxygenated and colourless when deoxygenated. If you google 'horseshoe crab blood', you'll see the blue-grey colour of hemocyanin. I've seen fresh squid blood and it is a similar, but paler, grey-blue.
Also have a look at copper-based minerals to get an idea of the range of possible colours of copper-based substances. The exact shade is about oxidation state of the copper and what other elements it is combined with, so some of the colours will be possible to duplicate in biological pigments. (Others may be hideously toxic!)
You can then get blue fur by these methods:
- Because copper is dangerous as well as beneficial. Your body needs it, even if you don't have hemocyanin blood. But it needs to regulate the amount very carefully, as excess copper is toxic. So what to do if with something that you can't survive without, yet will cause problems if you have too much just slopping about your body? Answer: store the excess somewhere that is biologically inert. Dump it into hair follicles to be incorporated into fur and hair. You can also put it into claws/nails. Then if you need some more copper, store less in the fur and/or lick yourself and don't spit up the hairballs.
- Having blue blood will alter the colour of your skin, unless the skin is chock full of pigment to hide the colour of the underlying blood. I'm white. If my blood turned blue/clear instead of its normal bright red/dark red, then my normal shade of pinkish-beige would be drastically altered. So if any of your animals have sparse fur, which lets the colour of the underlying skin show through, then you might be able to generate a range of other shades. This one, obviously, isn't blue fur as such, but is fur which looks blue, because the observer sees the combination of the blood pigment, skin pigment (if any) and fur pigment.
I'm hoping this doesn't violate your "not in the diet" rule, because they have the hemocyanin every day of their lives, not just on a special diet. So there is a dietary aspect, just like you will become anemic if you don't have enough iron in your diet.
EDIT: to include thoughts about individual variation in fur colour within a species.
We know from real world domestic mammals such as horses, rabbits, dogs and cats that a multitude of coat colours (and patterns) is possible. So mammals can do that. BUT mankind has interfered in the evolution of those species, and many of the coat colours only exist because when a mutation cropped up, some human went "Ooooh, pretty! Mum, I waaaaaaant one!" :-)
I've been racking my brains trying to come up with wild animals which have a lot of variation in fur colour (not pattern), and come up with... the brown bear (Ursus arctos). And, um, the black bear (Ursus americanus). Both of which can be a range of shades from a creamy white to a very dark brown (brown bear) or black (black bear).
This lack of colour variants in most species is a simple one: if you stand out from the crowd or from the background, you die. Especially if you are prey and what is hunting you has fantastic colour vision (birds of prey or people); or if you are the predator and you stand out like a sore thumb. These Kermode white bears are better at catching salmon than their black comrades, because the salmon can see the black bears more clearly.
However, the the OP mentioned Zootopia in comments as an inspiration. If his critters are intelligent and civilised, all bets are off. If civilisation, even a stone age one, has reduced your change of being nailed by a predator if you stand out, or has let you invent sneaky ways of catching prey (like fishing nets), then what colour your fur is becomes a lot less important. Local variation, family variation and individual variation can become the norm. If some colours are considered sexy, or lucky, or religiously significant, then bears with those fur colours may become more common.
Fur colour variation in individuals can happen for this reason: many coat colours, like black, are not created from a single pigment. Black pigments are expensive to make, so a black animal's blackness is actually a combination of various black, brown, red and yellow pigments. The black animal has the genes to make all of these pigments. Some of its descendants inherit some of the genes but not the others, so the descendants can make hair colour all of the natural shades of human or bear hair. This is called polymorphism.
The same could be true for the copper-based pigments: Lots of genes for lots of different ones.
So you could have a population of alternative-world bears who have collectively:
- Up to 10 different 'normal' pigment genes to make fur from albino (none) to black (all 10 pigments).
- Up to 10 different 'copper' pigment genes to make fur from albino (none) to intense blue-green (all 10).
Any individual animal will have inherited some of these 20 genes from its parents, perhaps all of them. Some genes will be dominant, others recessive, others will show incomplete dominance (blend the colour effect). Some genes will be common, others will be rare. The mix of who gets what will result in a range of coat colours in the alternative-world bears.
Plus you can get other effects like roan animals which have white and another colour of hairs mixed together.
There is however the issue of what the animals themselves can see. Humans and other primates are unusual in that we have good colour vision. Most mammals - particularly nocturnal species - have limited colour vision. Horses, for instance are fantastic at distinguishing different shades of yellow and green, so will be able to spot subtle differences in just how 'cream', 'beige' or 'yellow' a palomino horse is. But they might be utterly rubbish at telling shades of chestnut, bay and brown apart, and thus also bad at telling apart fantasy horse colours like shades of purple.