Thanks for the link, Peter, and for linking to my grandfather as the Greek example - he would appreciate it. I see how my question still wasn't clear, so I edited it again to clarify that it is a problem with searching, not data entry. We can enter anything in Greek (or any other characters or alphabet) in the biography section of the profile. We can also enter data in the name fields using another alphabet. But we can't search for these profiles using both the first and last names - we'll get no results. We can search by last name only , and all first names will appear as ????.
I used your link for a Russian example - Виктор Житомирский. Go to the search fields at the top of the page and cut & paste the characters into the last name field. You'll see that the first name fields for all six records have ????. Cut & paste the characters into the first and last name fields, and the search result you get is "Sorry, no matches were found for "Виктор Житомирский"." But his profile clearly is there.
For a Chinese example, I used 黄象郎 Unknown. Cut & paste the characters into the first name field. The search result you get is "Sorry, no matches were found for "黄象郎 Unknown"." There is no way to search for these Chinese profiles.
I noticed that the Hindi example only used that language on his personal profile page, and not in the family profiles.
Maybe we could allow searching for various alphabets work on the first name field the same way it does for the last name field?