Originally the idea was rooted in magic. The working of spells for good and evil is the chief usage of witchcraft. One is especially susceptible to such spells on his birthday, as one's personal spirits are about at that time. Dreams dreamed on the birthday eve should be remembered, for they are predictions of the future brought by the guardian spirits which hover over one's bed on the birthday eve.
The traditional birthday cake and candles also have their origin in ancient Pagan idol worship. The ancients believed that the fire of candles had magical properties. They offered prayers and made wishes to be carried to the gods on the flames of the candles. Thus we still have the widely practiced birthday custom of making a wish, then blowing out the candles.
The Greeks celebrated the birthday of their moon Goddess, Artemis, with cakes adorned with lighted candles.
Since it was believed that the positions of the stars at the time of birth influenced a child's future, astrological horoscopes came into being, purporting to foretell the future, based on the time of birth. 'Birthdays are intimately linked with the stars, since without the calendar, no one could tell when to celebrate his birthday. They are also indebted to the stars in another way, for in early days the chief importance of birthday records was to enable the astrologers to chart horoscopes.
The Greeks believed that everyone had a protective spirit or daemon who attended his birth and watched over him in life. This spirit had a mystic relation with the god on whose birthday the individual was born. The Romans also subscribed to this idea.
This notion was carried down in human belief and is reflected in the guardian angel, the fairy godmother and the patron saint.
I've also seen mention made of gifts for the birthday celebrant likened to sacrifices and offerings given to deities.
I suppose, instead of christians asking if it's appropriate to celebrate a birthday in a Pagan way, Pagans should be asking: Should we celebrate a personal birthday in the same way we celebrate our Gods? Are we engaging in hubris and elevating ourselves to the status of deity? Are we celebrating the divine spark within and therefore it is justified?
I'm obviously cool with celebrating a personal birthday, so I pick the later ;)