Yes but. From PCGS site:
PCGS guarantees that all coins submitted to it shall be graded in accordance with the PCGS grading standards and under the procedures of PCGS.
In the event the purchaser of a PCGS graded coin believes that the coin has been overgraded with respect to such standards and procedures, or is non-authentic, he may submit such coin to PCGS through the PCGS "Guarantee Resubmission" procedures and PCGS will re-examine coin to determine the coin's grade and authenticity.
The(shall be graded in accordance with the PCGS grading standards) means their standards, no others. I have been buying slabbed coins since PCGS and ANACS started slabbing. A lot has happened since then. But, all in all, I would prefer to buy a PCGS graded coin over any other. Remember, this grading was created as a market for sight unseen coins, the thought was you could buy a coin from anyone and it would meet PCGS or whoever's grading standards, this was pre-eBay. The theory was fine, but I have an example of a MS-66
Roosevelt dime that looks like the toning was due to someone sneezing on it. I have never resold it, because I did not want to stick someone else with that ugly piece of dung. I know this does not answer your question, but you really want to buy the coin, not the grade. The grade is important, but not nearly as important as the eye appeal. You will find MS63 Morgans with beautiful toning, that will sell for 3-4 time s more than the others in that grade.
To further answer your question, do not expect that a coin graded by ANACS will grade the same with other grading companies. The guarantee is only by the grading standards of the company you submit to.