Greece didn't claim the country for itself, it claimed the name for it's own northern province.
Macedonia is part of the Greek identity. FYROM (still the official name until the agreement gets implemented in a few months) have been claiming this Greek identity by misappropriating history.
Think of it as a trademark violation.
The country of Macedonia was part of a region generally called Macedonia for over a thousand years, as the initial named unit of Macedonia (which was in Greece) expanded to cover a wider area.
Besides I think the people in Greece-Macedonia are happy being Greek.