Cats, dogs or other pets don't know the science of diseases!

 Another possibility is that the best sense of smell can detect the faintest whiff. By this measure, it’s hard to beat a male silk moth, which can sniff out a female’s sexual pheromone at a concentration of one molecule per 100 quadrillion (that’s a one followed by 17 zeros) air molecules. And a moth doesn’t even have a nose. How does it smell then? It is because of receptor proteins on its antennae.

The size of the olfactory bulb might also be a good measure of an animal’s sense of smell. Among birds, they don’t get any bigger than those of turkey vultures, which rely on scent to detect carcasses from high in the air.

Bears, too, are well endowed in this respect, and there is evidence that polar bears can detect seal breathing holes from 3 km away. No clear winner then for animal with the best sense of smell (1).

• Recent studies have found that the African elephant has the strongest sense of smell in the animal kingdom.
• A bloodhound is often referred to as a nose attached to a dog since these pooches have the amazing ability to scan a terrain with their nose.
• The male silk moth's large, elaborate, feathery antennae are filled with scent receptors that enable them to sense a single pheromone scent particle from a female more than seven miles away. 

Some scientists say elephants have the highest smell sense because they have the most genes devoted to smelling.

The study behind that article from 2014: http://genome.cshlp.org/content/24/9/1485.full, unfortunately did not list bear as one of the animals studied.

Image source: Genome research

Examples of some animals who have high sense of smell:

• Cats - Cats have an excellent sense of smell and use it for hunting, navigation, and social communication.
• Pigs - Pigs are known for their exceptional sense of smell, which they use to find food and detect things like truffles underground.
• Bears - Bears have an extremely sensitive sense of smell that they use to locate food sources, identify other bears, and track prey.
• Elephants - Elephants have what is considered the best sense of smell among mammals, with over 2,000 genes devoted to smell receptors.
• Bloodhounds - Bloodhounds are known for having the best sense of smell among domestic dogs, with over 300 million olfactory receptors.
• Sharks - Sharks have an acute sense of smell that they use to detect prey from great distances in the ocean.
• Snakes - Many snake species, such as the pit viper, have highly sensitive forked tongues that allow them to "smell" their environments.

While dogs are renowned for their sense of smell, many other animals including cats, pigs, bears, elephants, bloodhounds, sharks, and snakes also possess exceptional olfactory abilities.

Image source: Pixabay

But cats don't come under these very highly rated animals with regard to smell. 

People say pet dogs and cats can even identify that their female owners are pregnant!

Earlier, we too owned some cats for several years. I didn't notice anything specific about their sense of smell with regard to human beings and their diseases despite my parents' health conditions and my sister's pregnancy.

Yes, cats can find food, mates, enemies using smell, I noticed this. 

Right, cats can sense some chemical changes that occur in a diseased system because of altered scent, like some other animals. But they really don’t know the "science behind the diseases". Therefore, they cannot realize their owner is sick in what way. Sensing the change in odour is not sensing the disease. Get that right first. (3)

If disease or pregnancy related  changes have you riding the emotional roller coaster, your pet is probably noticing it too. But It really cannot phantom the reason for it. It might change its response to your behaviour.

Any positive behavioural changes you say happen is just your wishful thinking, strange perception and attributing your thoughts to dogs, cats  and other pets! That’s all! (3)

Even though the stories told by pet owners sound interesting , they  can't be completely true, but attract people because of the emotions involved with a pet ,  its 'perception and prediction' of  disease or health condition of its owner  and strange interpretations of love and understanding.

I loved my cats and know how far they can go with regard to smells but have scientific knowledge enough to say they can't identify specific diseases because they don't know the science of it. They just know that there is a change in odour and show some attentiveness to it. That's it!

Don't attribute 'specific  knowledge' to this curiosity of pets, especially if they are not 'trained' in some way. 

Footnotes:

1. https://www.discoverwildlife.com/animal-facts/which-animal-has-the-best-sense-of-smell