Does an inherited annuity for which I received a 1099-R count as an IRA?

If the annuity was in an IRA before you inherited it, then yes it is an inherited IRA. If it was just a regular annuity, then it is just retirement income.

I don't know what it was before the person passed, unfortunately. 

If it was an IRA it would say so, otherwise it is just an ordinary annuity.

It's only an IRA if Form 1099-R provided by the payer has the IRA/SEP/SIMPLE box marked or has code T or Q in box 7.  If the code in box 7 includes code D, the distribution is not from any kind of qualified retirement account.