How can I get a new browser session when opening a new tab or window on Firefox/Chrome?

In Chrome you can use private browsing to achieve the same.

Chrome - Wrench(tools) menu/open new window without history

In Firefox you can use IE tab extension, and have another tab rendered with an IE engine to achieve the same effect.

Here are some other options: http://www.computer-realm.net/managing-multiple-gmail-accounts-with-firefox/

As of Firefox 57 AKA Firefox Quantum, the Firefox Multi-Account Containers extension (developed by Mozilla) will allow you to use multiple accounts on the same website, with a different account per tab or window.

Under the hood, it separates website storage into tab-specific Containers. Cookies downloaded by one Container are not available to other Containers. With the Firefox Multi-Account Containers extension, you can...

• Sign in to two different accounts on the same site (for example, you could sign in to work email and home email in two different Container tabs.
• Keep different kinds of browsing far away from each other (for example, you might use one Container tab for managing your Checking Account and a different Container tab for searching for new songs by your favorite band)
• Avoid leaving social-network footprints all over the web (for example, you could use a Container tab for signing in to a social network, and use a different tab for visiting online news sites, keeping your social identity separate from tracking scripts on news sites)

After installing the Firefox Multi-Account Containers extension, click the Containers icon to edit your Containers. Change their colors, names, and icons. Long-click the new tab button to open a new Container tab.

Firefox versions prior to 57 can use:

Multifox is an extension that allows Firefox to connect to websites using different user names. Simultaneously!

For example, if you have multiple Gmail accounts, you can open them all at the same time. Each Firefox window, managed by Multifox, accesses an account without interfering each other

Each Multifox window is flagged with a number indicating the identity profile. Logins made in windows with different numbers are isolated.

Logins are preserved the same way they are in “regular” windows. Even if you close the window or quit Firefox.

The identity profile of each window is preserved when Firefox restores the session.

It can also easily switch between profiles:

I am not expert in this topic, I found some interesting thing while surfing in Internet and also see that it is not mentioned in any of the answers already posted. Hope it will be helpful for someone!

If you want to open completely new session using chrome in Windows. Enter below command in Command Prompt app in Windows after creating Chrome_dev_session directory in C drive

chrome.exe --user-data-dir="C:/Chrome_dev_session"

It is as clean as chrome window you opened for the first time after you installed it. It may be a bit laborious task to create directory and enter this command every time you want to open new session but has been very useful for me.

Extra Reading: I used below command to disable CORS (Cross Origin Resource Sharing) in my Computer while testing some feature:

chrome.exe --user-data-dir="C:/Chrome_dev_session" --disable-web-security