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.)

I was in a similar situation but in the other order: I reviewed a paper for Mathematical Reviews and then was asked by zbMath to review the same paper. I told zbMath about this and they agreed that it is not ideal but they asked me to write a review nevertheless and told me that they are fine with getting the same review as Mathematical Reviews. I asked MR for permission to do this and was told that they always grant it provided they are the first one to publish it. They asked me to make sure to tell the editors of zbMath that MR published the review first and that they would know how to handle it. It seems that this situation happens from time to time.

Note that this situation is different since Mathematical Reviews‘ terms are more permissive than zbMath‘s.