There are two ways to set this up. One is to link a patient to multiple records in the same table. A single table might list all Physiotherapists, GP's, surgeons, etc or you might have a different table for each.
I think you are using a different table for each and if so, you need to set up this relationship:
Patient::_fkPhysioID = Physio::__pkPhysioID
And then you should select a Physio from a list of existing Physios and click a button to add a new record for a new Physio when they are not found in the existing list.
The simplest list method is to set up a value list as a drop down list or pop up menu. So I suggest setting up that approach at least at first to confirm that everything is working. Later, you can investigate the two explortation files for which I will include download links if you want to explore ways to make a long list of values easier to work with in FileMaker.
The script to add a new Physio record might look like this:
Go to Layout ["Physio" (Phsyio) ]
Set Variable [$PhysioID ; Physio::__pkPhysioID ]
Go to Layout [origional layout ]
Set field [Patient::_fkPhsyioID ; $PhysioID ]
WIth the new record added you can then enter a name directly into a name field from the Physio table, or you can modify the above script to either use a custom dialog with an input field to ask for the name or have the script pause on the Physio layout in order for the user to enter that name into a field on that layout.
You might check out the Adventures in FileMaking series. Adventures 1 and 2. They are free to download.
Adventures in FileMaking #1 - Conditional Value Lists (includes details on how to set up a basic field based value list)
Adventures in FileMaking #2 - Enhanced Value Selection (what to do when a simple value list won't cut it.)