How can I count the total number of built-in functions in PHP?

You could use a core function to count the number of functions available in the core and installed extensions:

$funcs = get_defined_functions();
echo count($funcs['internal']);

Count them here:

PHP Function List

Here is a list of all the documented PHP functions. Click on any one of them to jump to that page in the manual...

PHP site shows a total of 5368 functions.

There are 5845 listed function here at PHP website