It's phonetic, and these are all living languages. All you need to know is what characters correspond to which phonemes and you need someone who can speak enough words of the language.

So the same way you tell written French from German today.

Surely the lack of a ß gives it away.

Going solely on lack of a ß would characterise swiss german as french.

Having spent a few months in Senegal, basically if it's on a sign it's probably Wolof, if it's in a Koran or religious school it might be Arabic (though many people can only read the Koran phonetically, they don't understand the language).

I guess the same way as when using the Latin script (which is the script you and I are writing in).

Through the phonetic. Here is a sentence in Wolof "No toudou?" Anyone who reads it and understands Wolof would know the language. I know because this is how we (I am Senegalese and Wolof) write to each other in WhatsApp.

You need to know the languages. And then you pick up certain words, features or ways spellings are done. Even ignoring the more specific letters and diacritics, it is somewhat guessable what language is written with latin alphabet. Even if there is many of them.

I can clearly see that you're writing in English and not Dutch, French, or Spanish.