Is there an official browser for SharePoint on-premises 2013? Where most of the options and features will work? Or there is not such an info? For example if I want to use the "Open with Explorer" option inside our document libraries, then I can only achieve this using IE-11.
their is no dedicated browser for the SharePoint rather they call it supported. As per my experience best browser is always IE.
If you follow this Official documentation they mentioned Microsoft Edge, Internet Explorer 11, Internet Explorer 10, Internet Explorer 9, Internet Explorer 8 but with certain limitation with IE 11 in Edge Mode as well as Edge browser required Dec 2015 CU for SharePoint.
I dont think you will find any browser that works best because from my experience each browser has some shortcomings when it comes to SharePoint.
If you try "Open with Explorer" in Edge it won't work there as well.
You can't force browsers to support features that aren't available. There are a lot of things that are reliant on ActiveX controls rather than plain browser compatibility.
According to me the best would be to use Chrome , feature wise it works best. Like there will be problems activating and disabling features in IE but not in Chrome.
For some things that Chrome cannot do (like open with explorer) unfortunately you will have to reply on IE.
See the Plan browser support in SharePoint 2013 TechNet article.
Like others have pointed out there is no great browser experience for SharePoint On-Prem but IE unfortunately is the best with the most OOB functionality.
You can also use chrome and use a chrome extension called IE Tab which can add a lot of the functions you lose when using Chrome vs IE. It does tend to be finicky every once in awhile but from its description: "Display web pages using IE within Chrome. Use Java, Silverlight, ActiveX, Sharepoint, and more. A "Top 10" extension since 2009!".
If you use IE Tab you can set it up to just use IE settings and options on just your SP site and subsites this way it doesnt convert all websites in Chrome to act like IE.
ETA: IE Tab allows you to use the 'Open with Explorer' option as well as every other option that I have tested with it so this is a decent alternative to IE11.