イギリス is also written 英吉利, which implies that it came from the Mandarin, not Portuguese.

The question is whether the Japanese got the word from the Chinese, and if so where the Chinese got it from, or whether the Japanese got the pronunciation independently and then noticed that the Chinese already had a kanji for that word and decided to use it.

China and Europe had been sending diplomatic and trade missions to each other for well over a millennium before there was any direct contact between the West and Japan (and also since long before England came into existence). It seems likely Japan's early knowledge of Europe, including England, came via the Chinese.

This may be an Ateji, using Chinese characters phonetically to represent loan words: https://en.wikipedia.org/wiki/Ateji

They're the same characters in Chinese, and pronounced differently, so I doubt your hypothesis in this case.

Elsewhere in the thread people are saying that loanwords got kanji arbitrarily assigned before katakana was in wide use, and it was it this phase of language development that the Japanese and Portuguese first interacted.

I don't know the etymology and don't have a Japanese input method handy, but "coffee" is another word occasionally written with kanji to be fancy.

Every country has a kanji version of its name. If you see a newspaper headline that says Rice-Buddha Relations at All Time Low, it means America (Rice) and France (Buddha) are squabbling.