Is the relative natural abundance of isotopes of an element the same everywhere?
The relative natural abundance of isotopes is not the same everywhere.  Depending upon what you mean by "everywhere", there are two cases to consider.


Extraterrestial


Dust from before the sun was formed (stardust, presolar grains) has a very different elemental and isotopic composition than that found on earth.  Depending where a star is in its life cycle different elements and isotopes will be created and destroyed, so it is not surprising that dust from different stars has a very different composition than what is found on earth. As Wikipedia notes, 


  In these materials, deviations from "natural abundance" are sometimes
  measured in factors of 100



Terrestial


When our solar system was forming, elements and isotopes were not uniformly distributed - close to uniform, but not exactly uniform.  This is due to processes like diffusion in which a mass-dependent fractionation occurs.  A heavier isotope won't travel as far as a lighter isotope in a given amount of time, so the resulting distribution of the two isotopes will not be exactly equal.  While these mass-dependent processes account for most of the isotopic variation in our solar system, mass-independent processes can play a smaller role.  It is also suggested that catastrophic events in nearby stars may have also influenced local elemental and isotopic homogeneity.  These "original" inhomogeneities will be further altered by a continuation of the mass-dependent (diffusion, bond-breaking and making, etc.) and -independent processes. Natural radioactive decay processes can also lead to changes in local isotope ratios.  For example, $\ce{^238U}$ ultimately decays to $\ce{^206Pb}$, so it wouldn't be surprising to find an altered lead isotope ratio (e.g. enriched in $\ce{^206Pb}$) around large uranium deposits.

This paper presents a study of the natural isotopic variation found on earth.  Here is a table from the paper that gives an idea of the range in variation.  As expected it is much smaller than the extraterrestial variation, but certainly not insignificant.  For boron and copper, the isotopic variation is on the order of parts per hundred, but more typically isotopic variation is on the order of parts per thousand.