This is most likely because you are not adding back in the original data from the field.  If only the new project is in the entry field of the Update record step it is basically replacing anything in there with the single new entry.  Update the step with a comma at the end of the dynamic entry already there and then add another entry that is the field from when the automation was triggered, which still “contains” the old data.  Then when the automation runs the new project is added to the previous list instead of replacing them.

Thank you. I did figure that part out and fixed it. However, it's still removing the data from other projects. For instance, if a user signs up for Project A, their information is in the field. If they sign up for Project B, their information is being removed from Project A and being put in Project B. The field is set for multiple records. They should be able to sign up for multiple projects.

Hmm, that should work fine!  I’ve created it here for you to compare against your own set up, and here’s a gif of it running:

I wasn’t sure how your base was set up though, so let me know if you’re still facing issues after this and I’ll see what I can do!