The distinct feature in the center of a cell that makes a eukaryote common to plants and animals different from prokaryote cells belonging to bacteria is the nucleus.

Here's why:

* Eukaryotic cells have a true nucleus, which is a membrane-bound organelle that contains the cell's genetic material (DNA) in the form of chromosomes. This nucleus is the control center of the cell, regulating all cellular activities.

* Prokaryotic cells, on the other hand, lack a membrane-bound nucleus. Their DNA is located in a region called the nucleoid, which is not enclosed by a membrane.

In summary:

* Eukaryotes: Have a nucleus, which houses their DNA.

* Prokaryotes: Lack a nucleus, with their DNA located in the nucleoid.

This difference in cell structure is one of the defining characteristics that separates prokaryotes from eukaryotes.