Getting error message “fatal internal error occurred” when trying to change Apple ID password, what does this mean?

That kind of technical issue is really one for Apple to answer.

To contact Apple, click here --> Choose your country or region - Official Apple Support

Select your country, then a product. If you don't see one that handles your issue then keep experimenting with selections until you reach one that gets you a chat session or a telephone call and get the representative to redirect you.

or:

Contact Apple for support and service by telephone --> Contact Apple for support and service - Apple Support

You may also just want to see if it is still and issue. Your 55 "me too" on your original post suggest it may have been a temporary error others were experiencing. If you still see it, check Apple - Support - System Status