How to difference a column name on the select statement which derive from diferent ancestor tables?
Something along these lines. I need to link to the same table more than once so I need to give unambiguous table aliases. I favor minimal character aliases but you'll be needing to reference the doctor person, DP, or the patient person, PP.
SELECT
    DP.Name AS DOCTORNAME
,   PP.Name AS PATIENTNAME
FROM
    Register AS R
    INNER JOIN
        Doctor AS D
        ON R.Doctor_idDoctor = D.idDoctor
    INNER JOIN
        Person AS DP
        ON D.Person_idPerson = DP.idPerson
    INNER JOIN
        Patient AS P
        ON R.Patient_idPatient = P.idPatient
    INNER JOIN
        Person AS PP
        ON P.Person_idPerson = PP.idPerson;