It is believed that there is a significance to one's Hebrew calender Birthday.
Is there any significance for one's birthday on other calendars (such as the Gregorian calendar, as well as others)?
It is technically the day you were born, but would that have significance at all?
This answer is partially speculation; however, I feel that it is well-supported.
When you refer to a person's "secular birthday," I assume that you refer to his or her birthday on the Gregorian calendar since that is the calendar which most of the world uses. There are, however, plenty of other civil calendars used around the world (e.g. the Ethiopian calendar and the Iranian calendar). In addition, the Gregorian calendar has only been in use since the late 16th century (already into the period of the Acharonim!). I see no reason why the Gregorian calendar should have any particular significance in Judaism. The fact that something happened on some date in some random calendar system is irrelevant. This applies to all calendars--not just the Gregorian calendar. I can't just make up a calendar that has a three-day year and say that I want a birthday party every three days!
Furthermore, there is no reason why the Hebrew calendar cannot be used for "secular" purposes. In fact, the State of Israel requires that the Hebrew date be included in all official government documents.
On the other hand, the Talmud does recognize the concept of a 365-day solar year. The Gregorian calendar is the most accurate representation of the solar year around today. It is possible that there could be some mystical associations with particular solar dates although that is outside my realm of knowledge. Still, even if there is some significance to solar dates, that would still leave the Gregorian calendar as the sole exception to the rule that non-Jewish calendars are irrelevant.
There are many astrological systems which ALL play important roles in the world - after all that is why they were created. Some have more of an effect on the lives of man than others, and even within the zodiacs that do, there are tiers of nations and within nations, people of who they are allowed to effect and to what extent.
The Jewish calander is a lunar-solar calander, which mainly uses the monthly moon cycle, however due to a verse on the Torah commanding that Pesach always fall out in the spring, the solar calander with its set seasons plays a just as vital biblical role. As far as birthdays, everything age related follows after the "Hebrew birthday" fully and not the secular birthday for all halachik differences. However two notes:
Also interesting to note that even before the news were commanded (first commandment ) to announce rosh chodesh each month, the Torah does recognize Pharaoh's birthday (presumably his Egyptian birthday). Since seemingly only jews are commanded to keep the calander which we have (after all it is of no relevance to non jews who need not keep any holiday or rules based on months or years s other than a yearly accomplishment of still being alive, having a birthday is nothing more than having a celebration. The only question left to be asked is if it is an issue of u'vechokoseihem lo seileichu
