To set unique session ID's, have you tried to set your session ID by multi-user sessions (i.e., add ?session_id=my_session_1 to your URL)? Otherwise, by default Prodigy creates timestamped based session ID's (hence why they're different).
However, this shouldn't affect either the input hash (which is created by any values of keys with "text", "image", "html", "input") or the task hash (which concatenates the input hash with a new hash for values from keys named "spans", "label", "options", "arcs")
Another possibility, here was an audio user who had issues with their hashing because they accidentally used the keyname "paths" (there are several others like this on the default ignore list). Low chance but definitely could be what's explaining this weird behavior.