You say: "But in the global-map there is no entry for
M-x, there are only entries for execute and menu."
That's not true. If you are just using
C-h v global-map then it's difficult to tell, unless you're good at reading the Lisp representation of a keymap.
Instead, load library
help-fns+.el (Help+) and use
C-h M-k global-map, to see a human-readable key listing for that keymap. There you'll see this entry, for
C-h M-k is bound to command
[Emacs 27 will also have such a command, based on the one in
help-fns+.el, but the one in
help-fns+.el is still a bit better, IMO. Some of the features of the
help-fns+.el definition weren't adopted for vanilla Emacs.]
There is, however, another keymap variable that contains the
execute-extended-command binding for
Meta keys are in fact on that keymap.
ESC followed by a key sequence is essentially the same as
META followed by that same key sequence. So
M-x is equivalent to
ESC is not a modifier key here; it's a prefix key.
When you see keys listed in
*Help* you'll often notice
ESC listed as a prefix key, and you'll see
ESC followed by some key sequence listed as a binding.
And you'll see in the value shown for
C-h v global-map an entry for
ESC-prefix, which means that ESC is a prefix key globally.