Holly abandons her cat in Spanish Harlem. She tells him he's a "tough guy" and to "beat it". She acts tough about leaving her cat behind until saying that she and the cat "did belong to each other." Then she states the recurring tragedy in her life which is, "not knowing what is yours until you've thrown it away."
In spite of Holly's attempts to be live a non-commital lifestyle in order to be 'free' and 'wild' she has only committed herself to a way of living in which she obliges herself to make costly sacrifices. Her decisions force her to commit to abandoning the things and people she becomes close to that is if she becomes close at all. This could explain the way people question her as a phony.
Her happiness may be a facade, just as the role she played when she told her cat to "fuck off" and "beat it." It's as though her life is one big role she's acting out; she is being tough to cover her vulnerability, maybe even to cover the traces of remorse she has for committing to the lifestyle she lives. Also, if she sees other people believing her free, wild and happy-go-lucky or Holly Go-Lightly facade then she can feel like maybe it is true. When she projects herself this way and other people believe it she is able to believe it herself.