I am not sure how people would define a benefit, but SP comes under the same umbrella as Child Benefit, Employment Support Allowance, Pension Credit, UC, DLA, PIP, and many many more, some of which can be based on NI contributions, or disability, or are means-tested. They are all administered by the DWP. Below is their own definition of their role.
www.gov.uk/government/organisations/department-for-work-pensions/about
If people are saying the the SP is not a benefit, it sounds as though they think there is a stigma attached to claiming a benefit, which there really should not be. SP is based on NI contributions or credits, but a shortfall can be made up by claiming Pension Credit.
Governments can change the rules, including their own, some without going through parliament, about who gets what, the amounts payable and the funding of them. Just as they can determine how the NHS is funded, or how much is given in overseas aid, or defence, or the arts, or justice and much more.
Although we are a democracy, we can only exercise that right infrequently. ( Unfortunately in my constituency, my vote won’t oust the current MP.)
Humza Yusaf's resignation speech