This is as much a linguistic question as a history one - a lot of it has to do with how we classify languages vs dialects.

The difference between a language and a dialect isn't really definite - it's all shads of grey. Generally, the only reason we call something a separate language is that someone started calling it a language and other people agreed. In academic linguistics, we tend to just go "yeah, okay" and use whatever terms the native speakers use to avoid an argument.

There isn't one Arabic as we think of it - it's a group of dialects some of which are more different than some languages are to each other. A good breakdown of it is here.

Basically, you've got Classical Arabic (which is the language of the Quran), Modern Standard Arabic (which is used in international business and some formal media) and Spoken "dialect" Arabic (used day-to-day). Someone from Morocco probably couldn't understand someone from Iraq or Sudan without either drifting into MSA or another language. That said, a significant number of Arabic speakers consider it one language even when they can't understand one another.

If we compare this to Classical and Medieval Latin, "Modern" Latin (with an expanded vocabulary that is used to a degree today) and then the Romance languages, we can see it's not that different, it's just that they're called languages by their speakers so we call them languages.

tldr: A lot of it has to do with the fact that there is no hard and fast difference between a language and a dialect. Arabic speakers say it's one language, so linguists respect that and talk about it in terms of dialects. Romance speakers say they speak different languages, so linguists respect that and talk about them in terms of languages.