Entirely Different Functions.php per Page?

functions.php will load automatically for every page load, but you could certainly move most of the content of your current functions.php to other files and use functions.php itself as a primarily a kind of switch.

Caveat: You may have to hook some of your switching functions to get access to certain data. For example, functions.php executes before the main query (which is why the pre_get_post hook works if called from there) so things dependent on the main query won’t be set correctly in functions.php.