Learn Farsi: هر دفعه همین است. - Every time it is the same.

Just like the word "every" in English, the Farsi word har (every/each) is always followed by a singular noun. That is why we use daf'e (time) and not its plural form.

Could I use words like vaqt or zaman for "time" in this sentence?

No, because Farsi uses different words for different meanings of "time". Daf'e (and its synonym bâr) refers to an instance or occurrence, like "next time" or "three times". Words like vaqt and zaman refer to time on the clock or time as an ongoing concept.

What is the difference between in and hamin here?

In simply means "this". Hamin is an emphatic word made by combining ham and in, meaning "this same" or "this exact one". Saying hamin ast literally means "it is this same way" or "it is the same".

How would this sentence sound in casual, spoken Farsi?

In everyday speech, the formal verb ast (is) usually shortens to an -e sound and attaches to the end of the previous word. A native speaker would pronounce this sentence as har daf'e hamin-e.