Learn Turkish: Anlaşılan bizi dinlemiyor, sadece varsayıyor. - Apparently he is not listening to us; he is just assuming.

Yes, they are very similar! Both express a logical inference based on observation. Demek ki means 'so it means that' or 'therefore', while anlaşılan translates to 'apparently' or 'it seems'. You could easily say Demek ki bizi dinlemiyor here with almost the exact same meaning.

How is the word anlaşılan built grammatically?

It comes from the verb anlamak (to understand). With passive suffixes it becomes anlaşılmak (to be understood). Finally, the subject participle suffix -an is added to make anlaşılan, which literally means 'that which is understood'. Over time, it has become a set adverb meaning 'apparently'.

In English we say 'listening TO us', so why does Turkish use the accusative bizi instead of the dative bize?

Unlike English, the Turkish verb dinlemek (to listen) takes a direct object rather than a directional one. You are directly 'listening' the person, so they must be in the accusative case. This is a common trap for English speakers, so always remember to pair the accusative case with dinlemek.

Where does the verb varsayıyor come from?