Excel using cell color to select data from another spreadsheet

Shotgunenator Without seeing the file it a bit of a guess how that formula should look like but you could build it as follows, where the words between <> refer to the relevant ranges that contain what's described.

=INDEX(<birthday_names>, MATCH(<calendar_date>,  <birthday_dates>, 0 )

And if you give these ranges meaningful names you can avoid including sheet names and direct/absolute references in the formulae. It will make them easier to write, read and maintain.

NOTE: This is only pick-up one name per date. If you have people sharing a birthday, consider putting their names in the same cell. Otherwise it becomes much more complicated, unless you are on MS365 or Excel 2021. Then you may the FILTER function.