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Quirk Genetics & Quirk Factors
This is more a further explanation of quirks as a whole in terms of genetics/heredity since it's pretty vague in canon: Quirk factors will remain as the term for the all-encompassing aspect of quirk expression, however there is a specific genetic mutation that allows for the generation of quirks within the person (quirkless people lack this gene entirely, as the quirk gene is dominant). This specific set of genetic code is also fairly unstable like the DNA in cancer cells, which would explain quirk evolution/awakenings beyond birth or natural growth (like how those with diabetes don't always have the onset upon birth and some do).
The quirk-destroying drug/erasing bullets that Kai Chisaki developed do not actually get rid of the quirk gene, only the factor. This means that even the perfected drug/bullets cannot erase the quirk entirely. If given enough time, the person's quirk will re-awaken. The only way a quirk can be completely removed is through All For One or other hyper-specific genetic alteration.
The Trigger drug still targets the quirk factor, but affects the quirk gene, mutating it slightly and "triggering" the rapid evolution and amplification of one's quirk.
Quirk Doomsday Theory
Quirk Doomsday Theory is true in this world. It took four generations for the first signs of the theory to show themselves, and it will likely take another three generations after Izuku’s. By that time, the instability of the quirk gene may alter nearby genetic code and give future users a minor durability buff. To use an analogy like Dr. Kyudai Garaki’s, this minor evolution may be an attempt to upgrade the hardware to keep up with the software, though likely too little, too late.
Quirk Types
The types of quirks (emitter, mutant, and transformation/composite along with the accumulation subtype) in canon are all separate, and any physical alterations of emitter quirks like Anivoice and Acid are just random mutations. In my multiverse, the quirk types can "merge." For example, Koda and Ashido would have mutant-emitter quirks, Izuku's is accumulation-emitter (though One For All is fundamentally different, which will be explained later), Uraraka would be mutant-emitter (her finger pads would count as a mutation due to their abnormality compared to anyone else), etc. These instances are still one single quirk, just with multiple attributes that allow for easier usage or random mutations due to genetic instability.
Multiple Quirks
Most individuals are only capable of having one quirk at a time, with any attempt to add in a second quirk rendering them severely (or irreparably) damaged in their body and/or mind. More often than not, intense physical alterations and/or genetic modification are required to force the body to use multiple quirks. In rare instances, if a quirk that is naturally compatible with one's currently existing quirk (like Gigantomachia's quirks or say Dark Shadow and Black), they can be put into the same body with little repercussions.
All For One's quirk allows him to hold almost unlimited quirks with zero repercussions. One For All is a case where two quirks do not fuse but are compatible. The quirk that can be passed on hooks into the individual's personality and any/all quirks, passing them down. When the accumulation quirk was given, it was passed down as well and allowed for the passing of the first user's strength. This second quirk passively amplifies all quirks and strengths as it's passed down further. Rather than the quirks and strength and vestiges all merging, it's more like a continuing chain that leaves the earliest quirks stronger than the latest (i.e. Blackwhip is stronger in comparison to Float because it was acquired and amplified first).
The Todoroki quirk marriage is another case of multiple quirks within one body. This time, however, the quirks are opposites. Every Todoroki child's genetics are trying to merge the two quirks and failing in one way or another that leads to an imbalance or a "muting" of one side (the middle two likely have their fire sides muted, and Touya has the fire emission but the ice physical temperament). Shouto is the only one of the four where the quirk gene mutated just enough to allow both quirks to coexist equally and as strong as their parents' quirks.