Before wondering about something as complex as Dovid Hamelech’s songs, why don’t you even know which Shevet you are from? How much time does it take for a father to tell his son, “by the way, we from the Shevet of Asher”? Even with all the Tzaros, I don’t think it’s that hard to tell your son which Shevet he is. Someone once suggested something similar to what yitayningwut said. It’s hard to grasp actually, but something went on that we don’t know about.
We find that the Gemara in Pesachim mentions about two people who were into things that were of a stereotype of a certain Shevet. They did research and found that they actually came from that Shevet. So, we see that at the time of Chazal, it was already not known. We also see that when interested, they were still able to find out. Question is, why didn’t they?