The conventional wisdom on retirement is that you can draw 4% of your nest egg in your fist year of retirement and then draw that same amount each year, adjusted for inflation, and, if you do that, your retirement savings will almost definitely last 35 years (assuming a traditional asset mix).
So even those who are successful and have planned well for retirement may be on a fixed income, in that the amount is functionally set and will not change in any meaningful way on a regular basis.
So you could retire with say $5,000,000 and draw $200k/year from that and another $30k per year from social security, and, while you would be clearing almost $20k per month, you would be on a fixed income.
To your bigger point though, there is a connotation associated with the phrase that typically means that the party making the statement has not only a fixed, but also a limited, income.