The word footnote seems to only refer to the actual text that you are brought to if you follow the little number or symbol (such as '*') to the bottom of the page.
Does the actual superscript character have a name?
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Well, reference mark is in the dictionary to refer to the symbols:
any of various written or printed symbols, as an asterisk (*), dagger (†), or superscript number, used to indicate the presence of further information in a footnote, bibliography, or other text.
Based on the Random House Dictionary, © Random House, Inc. 2018.
Another similar phrase that is used is reference symbol.
The corresponding phrase for a number would be a reference number:
A footnote consists of two linked parts: the footnote reference number that appears in text, and the footnote text that appears at the bottom of the column.
Create footnotes in InDesign - Adobe Support