My understanding is that the current date at time at the center of the earth would be the same as it is at the surface, i.e. October 2019 as of the time of this post. By "at the same time", I mean by use of the Einstein clock synchronization convention, though since the one-way light travel time is about 21 milliseconds, the details of the clock synchronization are not particularly important.That makes sense. If calendars measured their respective proper times, and there were a 4.54 billion-year-old habitable lab at the center of the Earth, would the lab calendar, translated to Gregorian, show approximately March 2017? (October 2019 minus 2.5 years).
However, if two SI standardized clocks were started at an idealized event representing "the creation of the Earth", one clock in the center, and one on the surface, each clock would read a different number of seconds "now" (using the same Einstein convention as mentioned previously). If I haven't made a sign error, the clock at the center of the earth would read more seconds than the one on the surface.