Let's start with the two lineages of magic development in Frieren's world:
First is obviously the demon magic. They seem to have free-form magic that is given some structure through insane specialization and is basically powered by their autism. They have an innate talent for manipulating mana without the need for magic education, as parents don't bring up their spawn. But the end result is quite literally individualistic magic. Lügner can identify Zoltraak, and "basic attack magic" being similar to it, but can't seem to cast it himself.
The second kind is the elven magic system, who tends to follow some kind of structure and rules of logic. This is most likely in order to transfer knowledge and teach people magic spells. The human magic system has evolved from elven magic--quite literally through Serie teaching Flamme, and Flamme being allowed to research magic by the Empire, and thus being the foundation of the human magic system.
Humans have, due to their short lives and the need to teach each generation magic again, jumpstarted a quick rapid succession of magic evolution. Which created both new spells and improved old ones. This is exemplified by the evolution of Zoltraak.
This is however not how
the world works, only how the mages trigger effects they want to happen, and to make use of mana. You see the magic working, you see the caster, you know the spell and all that--but you don't really know how the physical phenomenon actually happens.
Ironically, it is literally the real world definition of Magick by Aleister Crowley:
Aleister Crowley defined Magick (spelled with a 'k' to distinguish it from stage magic) as "the Science and Art of causing Change to occur in conformity with Will".
So it is both Science and Art, with the Art part not being necessarily accessible through logic and reasoning. People copy what works, and refine what works. In effect, the underlying nature of the world seems to work like WH40K Ork logic, which is based on the WAAAGH! field. Basically a collective psychic field, where mass belief makes things happen. This is akin to what is called "egregore" In Western occultism, if you want another example.
The quite important part here is: You not only have to have the formulas, handwaving, mana manipulation, the staff, or other stuff: you also have to
believe yourself that the magic works, and
make others believe that it works. So it's the other way around: You use all those things to successfully make yourself and others believe, at least--or maybe fuidamentally--on a subconscious level, that things are going to happen, in order for the world to acknowledge that and really make it happen. It's a bit like Rupert Sheldrake's "morphic field": It's easier for other people to run a mile under 4 minutes once someone has proven that it is possible to run a mile in under 4 minutes.
Fuckups in basic belief happen quite a lot in Frieren, as in "failing to believe in yourself", or making others believe. For example the fight between Richter and Kanne + Lawine. Richter lectures the girls that pure mana attacks and shields are no longer "state of the art", and physical attacks (like his signature earth manipulation magic) are superior. Only then to waver once there is water involved for Kanne to transition to a physical attack, and subsequently to not even use his signature spells to defend against the flood of water, but use mana shields against the physical attack from Kanne. Which is literally the opposite of what he said you should do earlier, and he subsequently fails to defend against it.
You can see this also in the fight with the Spiegel, where the Frieren clone at the end doesn't even have any arms anymore, but smashes Fern into the wall nonetheless. Fern even says to herself: "I can't even perceive this as an attack."
Which brings us the doppelgänger generation of the Spiegel. How does that even work when the clones don't even have a mind? My theory is that it works a bit like Linie, who duplicates the mana flow and manipulation of other warriors like Eisen to copy their movements. Note that this basically says that even warriors tend to use and manipulate mana, at least on a subconscious level. The Spiegel doppelgängers have no mind of their own, however the Spiegel itself seems to read the minds of everyone involved in order to create a doppelgänger, and subsequently "teach it" the stored mana manipulation techniques from the original. The clones look and act like the original, because the Spiegel can't differentiate what is needed and what not for the techniques to work from the mind of the source. So it clones everything, including the mage staff.
