Will completing a postdoc in any country except the USA (namely: Australia, China, Japan, France) be even recognized when applying for professor positions also around the world?
If you are achieving good research results inside or outside the US, your postdoc will be valuable for your academic career.
If you are not achieving good research results inside or outside the US, your postdoc may not be valuable for your academic career.
Even if you do your PhD or postdoc in a place nobody has heard of in a country where academia is not as well-developed as elsewhere — if you have good publications in highly ranked journals, you are in a good situation. It might be harder to achieve this when you're not in a well-established research group, and personally, I would be actually more impressed if someone achieved this while based in Bangui than while based in Toronto.
I read in the web that in the USA and in general, most developed countries, doctorates from anywhere else aren’t recognised, but only from their own country or from the USA.
This is false. Most doctorates from serious universities are recognised. There may be some exceptions; for example, I knew a PhD student from Russia who said her PhD was not recognised in Russia unless she translated her PhD thesis into Russian. Since she had no intention to find academic employment in Russia, this didn't matter to her.