Same review for Math Reviews and Zentralblatt
The Math Reviews Guide for Reviewers addresses this explicitly:


  Reviews in other publications. You may find yourself asked to write a review of the same item for another publication. In general, the community is best served by reviews written independently. If you have already been asked by another publication to review an item, please let us know and we will transfer it to another reviewer.


So, if you have already reviewed the article for Zbmath, you should decline to review it for MR.  I didn't find a similar rule on the Zbmath side, but I would say the converse should also hold.

Also, per your agreement with Zbmath, when you submit a review to them, you grant them "exclusive exploitation rights".  So if you submit the same review to MR, you're infringing Zbmath's copyright.  I doubt they will sue you over it, but this shows it's actually illegal as well as unwanted.

In view of this, you may want to contact MR and ask them to withdraw your previously submitted review.

Evidently, one should not assume that the MR editors will check who reviewed the article for Zbmath in order to avoid inviting the same person, since they apparently failed to do so in your case.  (Another possibility is that they did check, but your Zbmath review hadn't been posted yet.)