True there are some viruses that cause repeated episodes of local (in and around the eye in this case) inflammation and increased discharge
with no history of recent cat
contact, we might expect other symptoms in the case of viral diseases (2 eyes affected, sneezing, coughing, nasal discharge, fever, decreased appetite, etc). We have none of that. I am skeptical that a "cold" exists in this case.
It is more likely that: