The matter of Zakat is usually simple:
You have to pay Zakat if the amount reaches the nisab, and you have kept it aside for one hawl (a lunar year). There might be differences of opinion on whether it must be a full lunar year kept aside or whether one may use it and gain profit meanwhile and have it left at the end of this year!
Based on that and from some fatwas I heard -online streams available- there's no difference whether this is money you've gained by your own work or that of an intermediate or comes from welfare of the tax payer.
Also note that the fatwa #2411 here covers the situation of a farmer who has been given public aid -to support him- and he asked whether he can/must pay zakat and can do hajj from that as he isn't in need for it and didn't ask for it. And it was answered by yes he can perform hajj and should pay the zakat.
Nevertheless scholars say it is not recommended for a Muslim to take (or better ask for) welfare if one can save money from it, especially if this money reaches an amount that is eligible for zakat.
Note that indeed the state or government in shari'a doesn't need to pay for zakat nor is any private welfare money eligible for zakat for the organization (or the people) itself (themselves). As this money has no real owner so it turns back either to the state or the joint box of the organization. But this doesn't apply for those who receive it, as they don't need to pay it back unless they do something illegal they can be considered as the owners of that money.