A majority of linguists consider Serbocroatian to be one language, there are many distinct dialects (with different countries having different standards). The writing system is irrelevant, the writing system isn’t the language (this can be seen in Mongolian, Tibetan, Hindustani, Persian, Kazakh, previously Azerbaijani, and contemporary Chinese languages as well). Also you can write Serbian in Latin script (en.wikipedia.org/wiki/Romanization_of_Serbian)
They are no less mutually intelligible than what are considered different dialects of other languages. In fact as someone who can read Russian & Polish I can understand a good amount of written Serbocroatian with trouble (it’s a lot harder than reading something like Ukrainian due to linguistic distance), it’s significantly closer between Serbian & Croatian varieties. Often people on media/politics pretend not to understand the other though due to mutual hatred from nationalism.
I would like to spend a lot of time on the language one day, I haven’t done much besides read some from grammar books on it. I like it a lot.