I mean you know that the Internet is nowadays instantaneous because accessing websites, instant messaging, online phone calls, etc. are instantaneous …

Email isn’t really intended to be used as an instant messaging system. A lot of the time, email clients are configured to only check for new emails periodically. Email servers might be configured to have some delay to avoid using too many resources at once. At least for me, emails (e.g. password reminder emails) do usually arrive within a few seconds, but indeed not always as immediately as instant messages.