Is it not true that if I can see a particular font on a webpage, then that means that I've got that font on my computer?
Well, you've got that font for that browser. Are you viewing the "pasted in" text in that browser, or is it in a plain text editor or word processor? They may not know anything about the font called for by the web page and browser.
I'm using
this website to lookup bible passages in greek. Enter a 1 in the box after "Matthew" and you'll see the greek.
I'd like to copy/paste from it but when I paste it, it comes out in english letters.
Well, the web page itself is coded UTF-8. What are you pasting into? These "english" (actually, Latin) letters correspond to the Latin alphabet transliteration that the greekbible.com website uses, called "Beta Code". So they're not actually displaying UTF-8 codes (probably double byte) corresponding to the Greek letters, but rather, Latin letters. Looking at the source for the page of Matthew:1, I see "bibloV", which is the Beta Code for the Greek beta-iota-beta-lambda-omicron-terminal sigma. In the Symbol font, that would be rendered in the appropriate Greek letters. View the page source for the Greek text and see how this transliterated Latin alphabet text is displayed in the Symbol font. Don't worry about each word's Javascript link to a dictionary or whatever they use to display information about the word. In fact, all you need to do with the cut-and-pasted text is wrap it in <font face="Symbol">...</font> and it will display correctly as Greek.
On the other hand, if you want this to show as Greek in a word processor or flat file editor, that may be quite difficult.