If there is such known "Mitochondrial Eve", does it means that all the mitochondrial dna in everyone's body is same?
The mitochondrial genome has a nonzero mutation rate, though estimates differ according to method. Estimates are approximately 10^-8 per bp per year on average, and the mt genome is ~20,000 bp. So we can do some math and expect that many (or most) people experience heteroplasmy, which is to say that they carry various different mitochondria (in the same person).
Note also that you inherit multiple mitochondria from your mother, not just one.
So- no, not everyone has exactly the same mitochondria, even if we can trace them all back to a single hypothetical human.