How much right does the PhD student have to choose his own advisor?
I wouldn't necessarily say that the funding is what gives your department the right to assign you a supervisor, but more so the rules within the department. In my department, we don't follow a strict mentorship philosophy. This means that graduate students often have more than one mentor that they work with throughout the course of their graduate career. There are times when a student loses their advisor (for various reasons) and they are assigned one, and there isn't much a student can do to prevent losing their advisor. However, the norms of our department is that the student may attempt to change their advisor. In other words, students in my department also have little wiggle-room to prevent getting assigned an advisor, but may attempt to change their advisor because that's the rules of the department. From what you explained, it seems like your ex-department gives little right to the student. This is, in my opinion, probably more reflective of the rules and culture of that specific department than the university providing you funding.

The extent to which the situation will negatively affect your future chances of getting into another program will probably depend on how you frame the situation. Obviously, you have taken courses and have transcripts from your previous university that you will have to submit to any new program that you are applying to, and shouldn't try to hide the fact that you ever attended the university. You can probably frame the situation in a way to not place so much attention on the interpersonal clashes between you and the advisor. For instance, you mentioned that you were away from your wife and child for 2 years, seeing them only once per year. This is obviously tough for anyone, and you can discuss how you wanted to spend more time with your family and that also played a large role in your decision as well. You might still want to talk about the department issues you had, but frame the situation to your advantage by not outright focusing on demonizing the former advisor, department head, or department rules. Make that portion a byproduct of personal issues or discuss how it was the one piece that fell down that started the domino effect of other issues you already had, depending on which is true for your case (i.e., also don't make stuff up! Key is to be truthful but focus on matters that won't make you look like a bad investment in the eyes of admission reviewers).