Is it ever (GMT) the same date everywhere in the world?

No....see below link.

"The IDL and the moving point of midnight separate the two calendar days that are current somewhere on Earth. However, during a two-hour period between 10:00 and 11:59 (UTC) each day, three different calendar days are in use. This is because of daylight saving in the UTC+12 zone and the use of additional date-shifted time zones in areas east of the 180th meridian. These additional time zones result in the standard time and date in some communities being 24 or 25 hours different from the standard time and date in others."
https://en.wikipedia.org/wiki/International_Date_Line