Yes, arguably most non-Latin scripts do use a separate alphabet for specific purposes, namely Latin for many foreign words from languages that use the Latin alphabet.
That is, in a block of non-Latin-script text one can find words like YouTube. Some language standards have prescriptions against it, for example in Serbian one should write Jutub even in Latin, but in the real world it is happening in practically every language to some degree. It is most common for company names and product names, whereas person names and place names are nearly always transliterated.
I count iOS, WADA, Huffington Post...
(In contrast, Le Monde or The New York Times will never ever use a Cyrillic, Arabic or Chinese character for a company name. Arguably most non-Latin scripts do use Latin in this way, that is, they would throw in a word like YouTube whereas the NYTimes will never ever ever use a Cyrillic, Arabic or Chinese character. Nil passive knowledge of other alphabets is assumed in any context, they are strictly esoteric. The exception would be the Greek letters for maths. Generally even identifying other alphabets is not required, and plenty of Cyrillic is even pseudo-Cyrillic.)
Moreover product codes, URLs, hashtags, email adresses, computer code, keypads, for example building door codes, units, licence plates, stock market tickers are in Latin and rarely in, say, Georgian. The IPA is also in Latin. The larger languages, like Russian, Chinese, Japanese and Arabic, can shield some of their speakers from this a bit more, but not completely. As a consequence, standards for how to say the letters of the Latin alphabet have often evolved in each language, for example 'y' in the Russophone world is ígrek, from French.
It is also worth mentioning that many languages have multiple scripts and historically there were other combinations. Georgian has three, one is used more now, but one of the others can be used for titles, a bit like uppercasing or italics. A large portion of German was previously written in Fraktur, and before the Antiqua-Fraktur dispute Antiqua was often used for a few bits, eg Latin titles and Roman numerals, Antiqua being associated with Latin culture and Catholicism, in works that were otherwise in Fraktur. Today Fraktur is still used internationally in mathematics.