After having met the "real" Patch Adams a few years ago and hear about his amazing organization that attempts to bring joy and laughter to places fraught with tragedy, I feel honored to answer this question. Patch Adams does, in fact, exhibit all three types of love you mention in your question. Let's deal with them separately.
First, companionship. Donning nothing but a clown's nose, Patch is able to brighten the days and lives of sick children by visiting the children's ward of the hospital quite often. His silly, slapstick humor takes their mind off of their troubles as they await and recover from serious surgeries and (sometimes terminal) diseases. There are smiles all around when Patch enters the room, where only gloom and loneliness is apparent beforehand.
Second, romantic love. Although not part of the "real life" story of Patch Adams, the movie character of Patch has a very strong love for Carin Fisher (a definite cold fish at first). Patch woos Carin in the same way he gives the children in the ward companionship: through humor. Carin Fisher opens up through her ability to laugh at Patch, and she is finally feeling her own romance toward Patch blossom when her life is ended far too soon (by a highly disturbed patient she is attempting to help).
Finally, friendship. Some of the most moving scenes in the film in my opinion are between Patch Adams and the character of Bill Davis, a terminally ill cancer patient. For what seems to be a VERY long time, Bill is irate and even violent towards Patch's attempts at humor and cheer. However, little by little, Bill begins to adore Patch, confiding in him as a friend. Bill shares revelations about his family, his illness, and even his spirituality to Patch, ... and there is no more tearful scene than when Patch Adams sings his own version of "Blue Sky" as Bill passes on.
I'll admit, absolutely every time I hear the line, "Blue skies smiling at me, nothing but blue skies do I see," I always think of Patch Adams singing to Bill as one of the different types of love you mention in your question. Therefore, in conclusion, I need to mention a different "type" of love that is very important to both the real Patch Adams and the character in the movie: love of the poor. Both through his institute to help those without insurance and through his mission to bring humor to places of tragedy, Patch Adams makes his great love for humanity apparent.