In the sentence:
A Jose se le rompieron dos tazas
There is information about who broke two cups.
There is no information about the owner of the two cups.
Consider this:
Se
me rompieron ( I broke them, accidentally)
Se
te rompieron (You broke them, accidentally)
Se
le rompieron (He broke them, accidentally)
Without the se:
Rompí las tazas (I broke them)
Rompiste las tazas (You broke them)
Rompió las tazas (He broke them)
If we don't know who broke them:
Las tazas se rompieron.
None of these sentences tells anything about the owner of the cups.
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If we wanted to speak about
the owner of the cups, we could do it this way:
Rompí las tazas de Jose (I broke Jose's cups)
Rompiste las tazas de Jose (You broke Jose's cups)
Rompió las tazas de Jose (He -not Jose, but someone else- broke Jose's cups)
Jose rompió sus (propias) tazas (Jose broke his own cups ... in a rage, for example)
However, if those two cups belong to Jose and they were somehow very dear to him, we could replace the neutral "de Jose" by an affective "le" pronoun, which indicates that the cups belong to Jose, and those cups are important to him for some reason. So, if his cups were broken on purpose, like kind of a revenge, we could say:
(A Jose) Le rompí las tazas - I broke his "dear" cups
(A Jose) Le rompiste las tazas- You broke his "dear" cups
(A Jose) Le rompió las tazas- He broke his "dear" cups
(A Jose) Le rompieron las tazas - Someone broke his "dear" cups.
If Jose's cups got broken involuntarily, then:
Se me rompieron dos tazas de Jose- I accidentally broke two of Jose's cups
Se te rompieron dos tazas de Jose- You accidentally broke two of Jose's cups
Se le rompieron dos tazas de Jose - He accidentally broke two of Jose's cups
A Jose se le rompieron dos de sus tazas - Accidentally, Jose broke two of his own cups
Does the last one ring a bell? It's very similar to your original sentence, the only difference is that this one does indicate that Jose is the owner of the cups.
This is the way to express the owner of the cups, the accidental nuance, and who broke the cups in the same sentence in Spanish.
Spanish does not allow this construction:
Se me le rompieron las tazas

to express--> I (me) accidentally (se) broke his dear cups (le)
Finally, if the cups got broken by magic:
Se rompieron dos tazas de Jose.
Conclusion:
A Jose se le rompieron dos tazas
Owner of the cups: Unknown
"Breaker": Jose, indicated by the pronoun "le"
It was accidental, indicated by the pronoun "se"
I hope this helps.