I would like to run two separate Safaris at the same time. Each Safari having his own session, history, cookies, and so on.
I can do that with two users on my Mac.
But doing that in the same user session would be handier.
How can I do that ?
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The best way that I have found to do this is to download Safari Technology Preview. It is a free download and you don't need a developer account to use. It runs separately from native safari and has its own folders for cache, cookies, history, bookmarks, etc. It is perfect for what you need.
The only downside is that it is in beta so you may see minor issues. Although most of the beta features can be turned of the the developer menu.
From Apples website:
Run side-by-side with Safari. Safari Technology Preview is a standalone app that works side-by-side with the current version of Safari, so you can continue to use and reference the current release.
The Fluid app, like mentioned in the comments, kind of allows this. It is more of a "single page app" tool, but it does use Safari under the hood and I use it to stay signed into different accounts at the same time without having to resort to Chrome.
In macOS Sierra the free version of Fluid allows separate cookies storage, while previously it was a (cheap) paid for feature.
 Chrome is great but uses too much battery for all day use.
There is an app called SwitchUp that does this, however I think it doesn't play well with Safari extensions. Try and see for yourself.