You can stay on Corfu for as long as you like without a residents permit. It's in the EU.
Reasons to have one that I can think of are for running a business or for applying for a loan. There may be others.
Someone else mentioned storing the car for a while. The criteria is that the vehicle is only on Corfu for six months. If stopped you may be asked to prove that the vehicle was imported in the last six months via a ferry ticket for example. Be careful with that.
Also, the chap that runs Customs at the port has a memory like an elephant. he can even tell my friend the registration number of the car she came to Corfu in 25 years ago !!! I can't remember his name - Yiannis I think.