Its not out of the realm of possibility.
If you go back to the spirit section in Street Grimoire, they have more info on spirits and their 'general knowledge base'.
Past history in SR also helps back this up to some degree.... as previous editions speculated that spirits MAY (GM call) have a collective memory of events they witnessed while summoned.
(Note however, Legally in the UCAS, spirits can not be called on for evidence, statements, or any other legal proceedings..... so they are only of limited use to police)
Your issues are going to be 3 fold.
1: Is it actually the same spirit? (GM call, as spirits are still a huge grey area 60 years after the awakening)
2: How do you Identify said spirit? Auras are unique to everything, and a mage can remember auras he has assensed, but with no real way to record said aura after the fact. (There is the silvernitrate camera.... but did he have it with him last time he saw this mage with a spirit AND took a picture?) How does he share this aura knowledge with those ge wants to do the searching?
3: Spirits are NOT bound to tell the truth

They can lie if it suits them

And while there IS a way to track down individual spirits, its a very messy proceedure requiring an Astral metaplanes trip, AND requires you to know the Spirits true name. (Which is NOT Spirit Flunky #2, incase you were wondering

). Again, this is found in Street Grimoire.
But, as with anything this far 'off the beaten path' expect a lot of GM handwaving ;P