Zay writes a plain z and continues the Semitic consonant behind the Hebrew zayin, the Syriac zain and the Arabic zay. The name is the inherited one in all four.
Like ayn, this letter took over a second proto-Semitic consonant. The reconstruction gives a voiced interdental fricative, the sound of the th in this, beside the plain z. Arabic keeps the two apart, writing the interdental with dhal. Ge'ez merged them into zay, and so did Hebrew and Syriac. The pattern is now familiar across the fidel: where a Ge'ez letter and a Hebrew letter agree in having one sign for what Arabic writes with two, they usually agree because both branches made the same merger independently or inherited it from a shared stage.
Arabic is therefore the most conservative member of the family in its consonant inventory, which is a result worth stating carefully. It does not make Arabic older, and it does not make the Arabic script older, because the Arabic alphabet is one of the youngest in the family and had to add dots to a borrowed Aramaic signary to write the sounds it had kept. A language can preserve what a script has to be modified to record.
The practical consequence for a reader is that a Ge'ez word and its Arabic cognate can differ by a consonant without either being irregular. Where Arabic has dhal, Ge'ez will normally have zay, and the correspondence holds firmly enough to be used as a check when a proposed cognate looks doubtful on other grounds.