The term "script" is used in many different ways. Unicode defines "script" as
A collection of letters and other written signs used to represent
textual information in one or more writing systems. For example,
Russian is written with a subset of the Cyrillic script; Ukranian is
written with a different subset. The Japanese writing system uses
In their epic work on writing, Daniels & Bright 1996 simply say that "script" is the same as "writing system". The Oxford English Dictionary does not even hint that Arabic and Latin might be different "scripts" – they say "A type of writing or handwriting, and related senses" and "Any of various typefaces or fonts which imitate the appearance of handwriting, esp. cursive handwriting; lettering in any of these typefaces", which would make European Latin handwriting different from American Latin handwriting. Websters (online) simply refers one to "alphabet", which is technically wrong because Arabic is not an alphabet, except that they also define an alphabet as
a set of letters or other characters with which one or more languages
are written especially if arranged in a customary order
so actually they simply reject the linguist's specialized definition of "alphabet". In other words, it depends on how you define "script".
You can consult this page for the scripts recognized in Unicode (which is not necessary exhaustive). You can see that Shahmukhi is not included, nor is the Kurdish alphabet (not an abjad), instead they are subsumed under greater Arabic, although Unicode does recognize Arabic and Arabic Nasta'liq. Similarly, many languages of the former Soviet Union use a Cyrillic alphabet with letters that are not part of traditional Cyrillic, but have been subsumed under broader Unicode Cyrillic (§7.4) that covers all of the distinctive letter shapes used in Cyrillic Kurdish, Komi.
Accordingly, there is no such thing as Urdu script, Kurdish script or Persian script, there are (currently) just two Arabic scripts. This doesn't correspond to ordinary usage, where speakers of a given language tend to think in terms of the system of writing for their language, and they don't generally seek to find the broadest applicable concept – unless there is a reason to do so (as there can be with Arabic script).