Is Lepton Flavour Universality an accidental symmetry of the Standatd Model? If it is, why? How does it emerge from the Standard Model?
It is not accidental, at least in the sense that this term is used.
The accidental symmetry is the symmetry that is forced on you by the renormalizability - i.e. for a given field set you can't write the interaction that would violate such symmetry. On the other hand nonrenormalizable interactions or renormalizable interactions with extra fields (these two options are related) may violate symmetry. The example is the baryonic number conservation which is easily violated in many extensions of the standard model.
On the other hand the lepton universality is the property of the non-abelian gauge invariance. In qft the non-abelian gauge interaction is determined solely by the representation of the field and the coupling constant, common for all the fields. I.e. you can't have $SU(2)$ doublet interacting with $W$ boson 1% stronger than another doublet.
Of course this constraint is weaker for $U(1)$ interaction, such as hypercharge field. But if you change somewhat $Y$ for some lepton, this may ruin the gauge invariance of the mass terms and the triangle anomaly cancellation.
In the Standard Model of particle physics the three charged leptons are identical copies of each other, apart from mass differences, and the electroweak coupling of the gauge bosons to leptons is independent of the lepton flavour. This prediction is called lepton flavour universality (LFU) and is well tested. In tree level decays, any violation of LFU would be a clear sign of physics beyond the Standard Model
The standard model developed slowly over the last century by continually fitting the data, using axiomatic assumptions and testing them against data. One of these axiomatic assumptions was LFU .
LFU is not an accidental symmetry, it is the result of the assumed couplings within the standard model.
From the introduction in the link
In the Standard Model of particle physics (SM), the electroweak gauge bosons Z and W ± have identical couplings to all three lepton flavours. This means that branching fractions of decays involving different lepton families do not depend on lepton flavour but differ only by phase space and helicity-suppressed contributions