Wa 3alykum assalam
Before I attempt to share my views about your question, please read the article by brother Joseph to understand what Zakat is
[1].
To summarize zakat is something similar to what one would recognize in todays world as what we call Taxes.
As you will recall from the article which shares verse
41:6 (Part)-7"...and woe to the polytheists (41:6) Who give not the 'Zakat' and who are disbelievers in the Hereafter (41:7)"
That even the polytheists were obligated to pay the Zakat/tax. Which could be argued that everyone living under the same government/community/land is obligated to this due.
Think of this as something that was even present/in effect even before the Quran was revealed. So the term was not something new the Quran developed. This is shown by that even the Children of Israel and prophets were commanded in the Zakat. It is even possible that all governments needed a system of zakat/tax in order to maintain control not only governments that were under Islamic rulings had a system of Zakat/tax.
Regardless of the faith of the people or the constitution it seems like God mandated that individuals/businesses should pay their share of wealth to the communities they live in or to the government/kingdom/emperor that ruled them.
This was always obligatory and seems to be a due/right(haqq) that the governing body had over the people.
What I think is meant by obligatory when people mention it, is that the system or the governing body has the right to demand this due from the people. They are always allowed to take it.
The article explains in more detail.
What is
not obligatory by the governing people is to demand "sadaqa"/charity from the people as can be seen in verse 58:13. However in the sight of God, it is obligatory and one will be questioned concerning their wealth which 9:60 affirms that it is a duty imposed by God.
In Verse 9:60 God has defined where the sadaqat should be delivered to. It also seems like the prophet(pbuh) was also the one who was distributing the sadaqat 9:58.
I do not see a problem that Sadaqat can be a subset of Zakat(not the other way around), which the governing body would also control, keeping in mind that Zakat is compulsory and sadaqa is not according to verse 58:13.
58:13 "Do you fear that you will not give in Sadaqat before your consultation? So when you do not do it and God has turned to you (mercifully), then keep up prayer and pay Zakat and obey God and His Messenger; and God is Aware of what you do."
It seems like there would be situations where individuals will not be able to give sadaqa, while they are still not exempt from paying the Zakat.
Insha'Allah that helps.
[1] WHAT IS THE CONCEPT OF ZAKAT FROM THE QURAN?http://quransmessage.com/articles/zakah%20FM3.htmSalam