Fermilab's $g\!-\!2$ experiment: physics beyond the Standard Model? On April 7 of this year the results of the Fermilab's $g\!-\!2$ experiment were published, increasing the discrepancy with what the Standard Model predicts.
The combined results from Fermilab and Brookhaven National Laboratory show a difference with theory at a significance of 4.2 sigma, slightly under the 5 sigma (or standard deviations) that scientists require to claim a discovery.
Some questions to the experts:

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*Does this prove that Lepton Universality is not universal? How is leptonic universality affected?

*What's causing this discrepancy?

*Does the Standard Model of Particle Physics might need some major reworking?

At present, the  $g\!-\!2$  result is arguably the most compelling indicator of physics beyond the Standard Model and, at the very least, it represents a major constraint for theories such as supersymmetry or extra dimensions.
If necessary, please feel free to edit this question. I'm not particle physics / standard model expert, be nice.
 A: The very goal of particle physics experiments all through the past century is to extend the model describing the particle interactions, at present called the standard model. . As energy gets higher and accuracy of measurement lower over the years the model has been modified in order to fit the data and be predictive. Experiments are continually looking for effects beyond the standard model. It seems that with g-2 and lepton universality the magic 5 sigma for deciding on discrepancy with current model has been reached.( One has to wait for further experiments with independent instrumentation for confirmation it is not a measurement effect ,as the super luminary neutrinos proved to be).
The Cornell site arxiv is a first place to look for new theoretical proposals. It is not peer reviewed, but a certain standard is maintained, on mainstream physics directions. The first paper appearing on googling "arxiv g-2  lepton universality" is indicative of the theoretical interest, seen in the number of entries too.

In the light of the recent result of the Muon g-2 experiment and the update on the test of lepton flavour universality RK published by the LHCb collaboration, we systematically build and discuss a set of models with minimal field content that can simultaneously give: (i) a thermal Dark Matter candidate; (ii) large loop contributions to b→sℓℓ processes able to address RK and the other B anomalies; (iii) a natural solution to the muon g−2 discrepancy through chirally-enhanced contributions.

You ask:

Does this prove that Lepton Universality is not universal? How is leptonic universality affected?

It indicates that theories based on lepton universality should be modified

What's causing this discrepancy?

Theorists are working full time to propose various ways of cobining the two effects in one theory beyond the standard model.

Does the Standard Model of Particle Physics might need some major reworking?

If the five sigma is solid experimentally, yes.
