I'm using Ariel Rounded Bold for a text object.

but only after converting the text object to path all browsers displayed it properly, see attached image.

Is this s a must do always?

Is this (converting to path) something to do with all objects (e.g. rec, star  etc)? 

tx

It's nothing something specific to Inkscape or the svg format.

If you use a font and the target system / platform / browser does not have that exact font installed, the target system will try to choose the closest option.

Sometimes that option is not what you expect.

Only formats which 'embed' fonts can avoid this problem if text is being used. An example is Adobe PDF.

Object to path converts the font to an svg path, the font is discarded.

There are some very safe options such as 'sans-serif' which can work cross platform.

I checked that before, sorry for not mentioning, and my system is installed with that font...

anyway, more important for me that I now know converting all objects is not a must do 

tx

You could have a look a this https://www.w3schools.com/cssref/css_websafe_fonts.asp