of the categories you gave-- tradition, custom, discipline-- the answer is all of the above. It is a tradition (also called custom) in the Latin Church, and it is currently a discipline encoded in canon law.
It’s just a rule, and one that is not rigidly enforced. It’s possible for married Anglican ministers to obtain Catholic ordination. There have been 49 Anglicans ordained - I don’t know how many of them are married but I suspect most of them are. The Diocese of Savannah, GA recently ordained a married Anglican minister (Fr. Lucien Lindsey). My own Archdiocese has a married priest.
This is also possible for Eastern Orthodox priests (many of whom are married), but I’ve never heard of it actually happening.
If a married priest becomes a widower, he is not allowed to marry again.
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