In the UK at least, it is paid by the executors of the estate as part of discharging the estate, so it's definitely an estate (or "death") tax. I would expect the same is true in the US, but happy to be corrected.

Your reasoning about the allowances is backwards. If you lose a parent, an uncle and a grandparent all in one year, and get $22 million from each, then you have had a $66 million dollar "tax allowance". There's only one of you, but there are 3 deaths, so $22 million per death, ergo, a death tax. Conversely, if you work for 3 companies, you don't get to consider each income separately and pay only the bottom bracket of income tax for each of them; you pay the bracket that is appropriate for your total income, because it's an income tax, not a having-an-employee tax. (in the UK we also have a 'having-an-employee' tax, but that's handled separately).