We observe the 13 days of Safar because Rasool SAW was ill during these days, we also observe the 10 days of Muharram.
By observe I mean we do not travel, do have have birthday or wedding functions etc.
Some say we are following Jahiliya practices
I am not aware of any observance of “the 13 days of Safar” in the Shari`ah. The Prophet (upon him blessings and peace) married in Safar and he also often travelled in Safar.
Hajj Gibril Haddad