Earlier quoted context omitted.
Most competent websites I know accept general UTF8 characters like emoji perfectly fine. There are a lot of crappier websites that don't even have proper unicode support for usernames or profile descriptions out there, though, so your mileage may vary. As far as I know, there's nothing preventing a password field from containing any valid unicode string. The problem may be IME support or servers stuck in ASCII, but t…
Even surprisingly big names are surprisingly bad at this. Don't know recently, but Hotmail/Outlook used to have a rule of only using letters, numbers, and a handful of symbols, also limiting you to at most 16 characters or something. You couldn't even type a space!
Even good websites that will accept any valid password string will sometimes cut off the last part of a long password because their hashing algorithm throws that data away. Bcrypt, for example, supports a maximum input length between 50 and 72 bytes, depending on the library you use to hash your passwords. That's bytes, not characters!
More primitive systems used to have problems with non-alfanumerical passwords and once those algorithms have been unleashed upon the unsuspecting public, you need to support them in your login flow for years to come.