Katz or Catz is from an abbreviation for Kohen tzedek (כהן צדק), or “priest of righteousness.”
The acronym of those two words - כʺצ - spells “Katz.” (Remember that the gershayim ʺ indicate an abbreviation - see https://bethshalompgh.org/what-are-those-thingy-dos-in-that-abbreviation-originally-published-march-6-7-2020/ for a reminder.) And that is nearly always the derivation of the name among Jews of Ashkenazi descent.
There are those who opine this is one of the oldest surnames in history. In fact, Ripley’s Believe It Or Not! declared it such, in at least one edition. (We note that around the world a “surname” is not always the “last name.”)
[The Engrave.In website, from India, does an interesting job of reporting several very old names, including Katz, Smith, King, Priest, and O’Brien. They declare that O’Brien likely should receive the award for oldest, which they say is documented in the Annals of the Four Masters, tomes produced between 1632 and 1636 in Ireland mostly from older works.]
Some say “Kohen tzedek” is based upon Psalm 132:9, which says “Your priests will be clothed in righteousness.”
כֹּהֲנֶ֥יךָ יִלְבְּשׁוּ־צֶ֑דֶק וַחֲסִידֶ֥יךָ יְרַנֵּֽנוּ׃
(We note that a translation which we use is “Your priests are clothed in triumph, Your loyal ones sing for joy.”)
From 917-922 CE the Gaon of Pumbedita was Kohen Tzedek Kahana ben Joseph, and his son Nehemiah ben Kohen Tzedek would follow as Gaon from 960-968. There are those who consider his “Kohen Tzedek” to be a “surname,” and though your correspondent begs to differ, we report it for your consideration.
Those with the surname Katz or Catz generally, as Kohanim, would be descended from Aaron. There is one tradition which claims specific descent from Tzadok, the first high priest in the Temple (2 Samuel 15:24), with the speculation that the original name was Kohen Tzadok.
It is reported by one site (forebears.io) that the name is held by some 52,154 individuals in the United States (plus 334 who spell it “Catz”), 17,219 in Israel, 4,589 in Germany, and 2,769 in Canada, with others elsewhere.
When the name is purely German, it is likely from Katze, or “cat.” It is noted that the name of the Katz Castle along the Rhine River comes from Katzenelnbogen, derived from the Latin conglomerate word Cattimelibocus, from the ancient German tribal names Chatti and Melibokus. (Your correspondent thinks it really means “cat’s elbows.”)
Some sources report the name כʺצ on a tombstone in Prague dated 1536, and on one from 1618 in Frankfurt, in the books of the Soncino family of Prague in the 1600s, and in a preface to Shabbethai ben Meir haKohen’s notes on The Chosen Mishpat (1663). However old it is, it definitely has a unique derivation.