How are emails related to or linked with the domain?

First of all, don't confuse two different types of domains.

The term "domain controller" comes from Active Directory (AD). It's true that each AD domain is also a DNS domain, and each AD domain controller is also a DNS nameserver – but not the other way around.

Many DNS domains on the Internet have absolutely nothing to do with Active Directory, and most organizations' DNS nameservers are not AD domain controllers. The domain amazon.com is not AD-based at all, it is just a plain DNS domain.

(Even many corporations which do use AD, still keep it separate from their DNS and from their mail system. For example, you will often see that only a subdomain such as "ad.example.com" is AD-managed, while the rest of "example.com" is not.)

How are emails related to or linked with the domain?

At its core, email is only related to DNS domains (not AD domains). A domain that wants to receive mail will usually have MX records in its DNS nameservers. For example:

amazon.com.  MX  5 amazon-smtp.amazon.com.

All mail sent to @amazon.com – regardless of the user ID – is handled by the same servers. So each domain makes its own decisions about how user IDs are linked to whatever else – it depends entirely on the mail system that Amazon decided to use; it could be linked to employee accounts but it could be completely unrelated too.

If I type in www.amazon.com in the browser, then am I hitting the domain controller of amazon?

The question is based on incorrect assumptions. You can't be hitting the domain controller because the domain doesn't have a domain controller. DNS domains have nameservers.

The DNS nameserver is involved in reaching www.amazon.com, however, it does not actually serve the website directly. It only returns an A or AAAA record pointing at the web server's IP address. For example:

amazon.com.  A     205.251.242.103
amazon.com.  A     176.32.103.205
amazon.com.  AAAA  2001:db8::205

It's mail server & DNS settings.

Domain name amazon.com belongs to Amazon and Amazon DNS (A setting) points this domain name to an IP (in fact to a pool of IPs because it's a large company with a large traffic but this is out of scope).

Then Amazon has a mail server (a pool of ...). With its mail server it defines the email addresses (@amazon.com or @any_subdomain.amazon.com) and still with dns (MX setting) it points to the web server to use.

Have a look to the web "DNS records", especially A/AAAA and MX records.