Aurora is not MySQL.
Amazon has ripped out a lot of code and features from the storage engine, but they haven't disabled the corresponding functions or status variables or configuration variables.
There are a bunch of features familiar to MySQL users that are still usable on Aurora, but they have no effect. Like the steering wheel on one of those coin-operated kiddie rides.

Aurora does not provide complete documentation on which features are still functioning and which ones are "disconnected" in this way.
In the case of SQL_CALC_FOUND_ROWS
, it might still work, depending on your Aurora version. I found a reference to it in their release notes:
https://docs.aws.amazon.com/AmazonRDS/latest/AuroraUserGuide/AuroraMySQL.Updates.20170515.html
Fixed a regression, introduced in Aurora MySQL version 1.11, in which queries can return incorrect results when using the SQL_CALC_FOUND_ROWS option and invoking the FOUND_ROWS() function.
Is it possible you're using an Aurora instance that is affected by this bug, or is it possible that the bug has regressed again in the version you're using?
You need to double-check the available Aurora information, and if you can't find what you're looking for, contact AWS support.
And don't assume that something that works in MySQL will work the same way in Aurora. They are effectively different software products. MySQL is open-source, Aurora is not.