I thought about it for so long that I started getting confused.
Final conclusion, I added his/her/their because I think in Ukrainian people tend to drop the pronoun when the context is clear.
E.g. I would totally say "Я шукаю валізу" rather than "Я шукаю свою валізу" cause I'm lazy; I'm not very sure if it works in third person too (Він шукає валізу instead of Він шукає свою валізу) just because I thought of it too much and lost the feeling xD
Added it! Thanks a lot for commenting!
I think if you wanted to be annoyingly analytical about the sentence by bringing pragmatics (linguistics) into it, "the" probably is the better fit than "a" because of the subject. "The passenger" is probably not looking for a random suitcase but a previously established referent, known both by the speaker and recipient of the communication.