I'm an American and when I hear Americans use this phrase, it makes me think of feeling patriotism toward your own country. If a foreigner says this, it depends on the situation. It may or may not be an insult. I once hears a Chinese person that was eating a hamburger 🍔 fries 🍟 and soda 🥤 and they said they felt American but these foods are known to be the diet of many Americans.

They feel as though they are living as a American or British person. I hope that helps you understand.