Email as a communication medium is generally considered to be insecure; the core protocols were not designed with security in place, and while secure protocols have since been layered on top of it, the nature of email means it has to be designed to "fail open" - everyone expects their email service to be able to send to anyone with a valid email address, so if they don't support TLS then you can't use it. And of course there is the problem of forwarding: once you send me an email with sensitive data in it you have lost control of it. I can forward it or print it out.
Your question mostly comes at it from a slightly different perspective - what about the email mailbox as a secure place to store data. There's not really an answer to that one, as different pieces of email software store the mailbox differently. Most of the modern ones will use decent technology to secure the actual mailbox database. On the other hand, the most popular cloud email services all make their money by reading your email and using the information to sell targeted advertising, which is not good for security.
In general, email accounts get hacked a lot because a) those are one of the best things to attack - "I rob banks because that's where they keep the money" and b) because they are secured with passwords and humans are rubbish at passwords. It's not really that they are inherently insecure.
Email based password resets are not ideal, but customers find them convenient, and they are cheap at scale, so we're stuck with them.
