Swami Desikan in his Prabhandasaram had mentioned about the place, day, date and pasuram count, but he did not mention about their lineage. Is there a specific reason to this or is my understanding wrong?

Azhvars are incaranations of Nityasuris (immortals who does eternal service to Bhagavan) so, their lineage is not important. We should revere them and celebrate their Thirunakshatram grandly, hence Swami Desikan had included only such details (which are important for us) in Prabhandasaram.
Notes:
He has mentioned these details so that an Azhvar’s appearance day should be celebrated grandly every year as a festival. On that day we must recite their Prabhandam. Moreover, most of them appeared as ayonijas (not being born from a womb) knowing their lineage is not crucial.