It seemed the doctor believed it would take some time for Bourne to remember everything. If he was so sure about that then why did he let Bourne leave the boat so quickly? Also where was it where Bourne left the boat? It was definitely not Switzerland. The reason being he took a train to reach Switzerland from there. Also how could he ride the train without any VISA? The train must be some trans-European train and I heard an all-Europe VISA or a country based VISA sort of thing is needed to ride such train. So how could he manage to ride it?
I always assumed the "doctor" to be a regular member of the fisherman crew who just happens to know something about medics too. Seen in that light, the relationship between the two had little to do with a normal doctor-patient relation wherein the doctor could order the patient to stay.
In this case, Jason was eager to find out about himself and to go to Zurich. The doctor did not "let" him go. Jason just went, as he needed permission nor release card.
Also, the time spend aboard healed his physical condition. On mental, the only thing the doctor could do was wait and observe. Something both of them would not have had much pleasure in, I imagine.
On your second question, assuming the fishing boat moored somewhere in the south of France or Italy, obviously Europe's mediterranean anyway (and the filming locations say it is Imperia, Liguria, Italy): according to the Schengen Agreement, traveling between its European member countries is without internal border controls. So al he needed was a plain train ticket that can be bought with cash money. In most European countries, there are identification requirements though, but normally you have to provoke to get you asked to.
Without knowledge of traveling within Europe, it sure may need a little quessing on how exactly he got what he needed. The only hint the movie tells is the doctor saying:
With Jason not asking how to, but instead saying just "Thank you" for the money, and the resourcefulness he later on exhibits in the movie, we might simply imagine that he just managed.