That's very interesting.
How much of our personality / memories are not really stored in the brain, but elsewhere in our body ?
This raises questions about organ transplants.
Storage is a fascinating issue, but I doubt the enteric nervous system would store or process any of the information we find in the central nervous system including personality or personality-related memory.
Instead, we do know that the synaptic and hormonal communication between the brain and ENS allows for bi-directional influence. This is to say that no information about personality is expected to be stored in the ENS, but that signals from ENS and gut microbiome can influence the brain systems supporting the person-specific complex of thoughts, emotions, and behaviors known as "personality".
We see very short term behavioral, emotional, and cognitive changes when people get grumpy from hunger and everybody can remember feeling their stomach growl or feeling hunger pangs. We might consider these as "bottom up" signals from the digestive system to brain. Similarly, top-down signals also have an effect on digestive systems. Just thinking of a taking a bite of a big juicy burger, smelling and sipping a freshly brewed cup of coffee, or eating a hot cheesy slice of pepperoni pizza along with a cool refreshing drink may precipitate salivation or other responses from the digestive system.
When looking at the ENS, we also need to look at the gut microbiome. It is clear that sensations and activity within the enteric nervous system and the gut microbiome influence central nervous system function, including behaviors we would associate with personalty.
For a study about the influence of gut bacteria on different aspects of memory in rats, check out, "Long-term multi-species Lactobacillus and Bifidobacterium dietary supplement enhances memory and changes regional brain metabolites in middle-aged rats." (https://www.ncbi.nlm.nih.gov/pubmed/28602659).
If you like considering the location of memories, one of my favorite concepts in memory is the role of the "perineuronal net". There are now many articles on that topic. For the longest time memory research focused on neurons and neuronal information processing so it is good to see more attention paid to the structures outside of and around neurons.
It appears 90% of the body’s serotonin is produced in the gut! Which among other things suggests a possible causal mechanism for antidepressants causing weight gain.
Random article from Google search results: https://www.scientificamerican.com/article/gut-second-brain/