Does Mandarin pronounce its ‘R’s the same as English?

They are very similar and sometimes overlap.

The most common pronunciation of English /r/ is as a labialized postalveolar approximant, [ɹ̠ʷ]. Some North American, Irish, and West Country varieties have a labialized retroflex approximant, [ɻʷ].

The syllable-initial Mandarin sound transcribed as <r> in Pinyin is most commonly somewhere between a retroflex approximant, [ɻ], and a voiced retroflex fricative, [ʐ], syllable-initially and a rhotacized schwa, [ɚ], syllable-finally. This is the same R-coloured vowel that you find in the rhotic North American pronunciation of “fur”.