Example of an application that uses this check?

DOSBox emulates only part of the ANSI.SYS driver, specifically the processing of escape codes in console output, so it is important to make sure that implementing the install check won't actually cause problems when applications try to use other parts of the driver that aren't implemented.

Free Pascal Compiler (FPC) - possibility of coloured output has been added to the command-line compiler recently (trunk version) to make errors and warnings more visible. I extended this support to the DOS (GO32v2) version, but I cannot just throw out ansi escape codes without knowing that they wouldn't garble the output instead of making it clearer.

Since you are the developer, you could use a configuration option or command line switch to enable ANSI coloration.

I see you've already made this request of DOSBox-X: https://github.com/joncampbell123/dosbox-x/issues/1430

No, the request for DOSBox-X is completely unrelated, created by someone else and before the functionality was added to FPC. Adding special option just because of DOSBox missing the installation check doesn't sound appropriate. As you an see from the other request, the installation check is a standard solution. I understand your comment regarding just partial functionality provided in DOSBox. That is obviously up to your assessment whether the implemented subset is closer to the full functionality for even the first versions of ANSI.SYS, or to nothing. In any case, thanks for your response and for supporting DOSBox.