How is the algorithm used to generate the hashes protected? Is is possible it could be reproduced or stolen, and if someone could do that, would they be able to find a way in some cases to feed the hash directly into a system to bypass the security?

The hash algorithm is actually open source, as are the vast majority of things used by governments in this world (at least the governments of the factions I focus on). Validity of the data is ensured mainly by physically securing the scanner so that no erroneous or malicious data can be injected, and the scanner also signs the hashes it sends to other devices, so even if you had hardware access, you can't just unplug the scanner and plug in your own cable to spoof the output without the host device noticing (also the tamper detection would also go off which will make the entire system inoperative until it's reset by the authorities). It's obviously not perfect but it works well enough, and for high security applications, for example a border crossing, the scanner wouldn't be unattended either.

If this technology is used for all credentials such as drivers licences, passports, healthcare etc, then surely if someone were capable of spoofing your hash, the identity theft would be completely crushing. What would the implications of this happening be? As a result, are there any routes provided through the government or third party insurers which deal with such cases of identity theft, and what would that look like.

There aren't really private insurers and most things are provided by the government. The ID system is in part in place to ensure fair distribution of resources as well (basically used for their rationing system) so spoofing it would be very bad. If you think someone spoofed your DNA, you need to report it to your own taxonomic government immediately and they will have systems in place to deal with it, but it would still be a massive pain.

There was actually a major incident in this world where someone drugged a high ranking government researcher and basically used his entire body as the key for accessing various restricted computer systems and stole experimental science and technology for a rival power (he was a mouse and his captor was a cat so they were basically holding him like a key fob). It caused various government agencies to reconsider the idea of using DNA as the only source of authentication, it could have been prevented with a simple password prompt following the DNA scan.

More and more now they're also seeing dual-technology biometric systems where there are cameras around the scanner that make sure you look like the animal that the DNA hash belongs to. Usually some form of facial and fur-pattern recognition, but at the very least they check if you're the right species and there is only one animal at the scanner at a time. As well as the aforementioned password prompt for security critical things.

How much data is this and how is it stored? Depending on the population of your world, even if you were not storing the DNA sequences, how large are the hashes, and how many of them are stored. How many data centres would be required to store this?

Since a lot of animals in this world are tiny, their total population is far higher than ours, and the hash itself is only a tiny part of the data that your taxonomic government has on you. However, they're also more advanced than us so storage is no problem for them (actually, a major plotpoint is that this takes place millions of years after humans mysteriously disappeared from the planet, and they have recently surpassed humanity in terms of scientific and technological advancement).

What political implications exist as a result? Surely there would be individuals or even political entities that are against such technology. What do these factions look like, and what else do they believe in?

There definitely are animals that oppose such a system. You have your typical "sovereign citizen" types just like we have that hate any form of ID system, but you also have animals and factions that don't subscribe to the whole "predator and prey living in harmony" thing, and since DNA evidence (along with scent evidence) are most commonly used to enforce the predation ban in the factions that do ban it, the ones that encourage predation infamously refuse to give DNA information on their citizens to those governments and will not look up who the DNA belongs to even if you collect it from a predation crime scene. This is actually why the anti-predation governments don't allow any animal from a pro-predation government to cross into their territory, the main reason is they might eat someone, but the reason/excuse that border agents most commonly give is that they can't look up their DNA hash and therefore they effectively have no documentation.

There are of course also the animals that think "the DNA scanners will give me paw cancer!" because it shoots an energy beam specifically at your DNA. Even though there's no evidence that they damage the DNA, and they're very specifically designed not to.

How accurate is the hashing algorithm? Identical twins can share the vast majority of their DNA sequence, but something like 15% have a “substantial” difference as they acquire mutations when in the womb. Depending on the accuracy of the algorithm, is it possible it could confuse one twin for another?

Given your world is science-fantasy, if anyone has invented cloning, depending on how they are doing it, it could have enormous implications here as well.

It uses the exact sequence, but it does also account for random mutations in individual cells by scanning many cells and comparing them, so individual mutations can be corrected for since each individual change would only show up in a minority of cells.

They have not developed cloning yet, but fully identical twins whose DNA is the same are flagged in the ID system. Typically each twin would also have a second factor like a password to tell them apart. Same for the extremely rare cases of a hash collision where two different animals with different DNA produce the same hash. Though practically I imagine it would still be at least slightly a pain to be a twin in this world since your government ID is more complicated.

I think that a system like this is vastly superior when it comes to convenience, but could have major security, financial, technical, ethical, and political implications.

Honestly that's what I want to go for. I like my worldbuilding to explore the less obvious or less "exciting" problems that the fictional premise brings which too often get glossed over in "brand name" media.

Thank you!

Good point. I tended to handwave that away with the reasoning that the device scans many cells at once and compare their genomes to cancel out individual mutations, but I'm also not very knowledgeable of what the somatic cell mutation rates are or if they would completely swamp the original genome in especially an elderly animal.