Yes! That interpretation is correct! As long as the studies are owned / created by the same researcher, the participant ID of the same participant will always be the same across them.
Just to be clear, for a multi-part study, researchers would commonly need to set up multiple "studies" on FF. When launching the Day-2 study, you can select the Day-1 study as the prerequisite. This allows researchers to launch Day-1 and Day-2 studies at the same time, without worrying about the correct order participants need to take them. This is because participants can only see Day-1 first, as they are ineligible for Day-2. Once they finish Day-1, Day-2 shows up in their feed because now they are eligible.
Just to be clear, for a multi-part study, researchers would commonly need to set up multiple "studies" on FF. When launching the Day-2 study, you can select the Day-1 study as the prerequisite. This allows researchers to launch Day-1 and Day-2 studies at the same time, without worrying about the correct order participants need to take them. This is because participants can only see Day-1 first, as they are ineligible for Day-2. Once they finish Day-1, Day-2 shows up in their feed because now they are eligible.