I might be out on a limb here, but I feel like you might be suggesting this is indicative of something - that is, “this happens more than it should, therefore…” Am I right? What’s the conclusion you’re pursuing?
My brother’s phone number (with area code - i.e. 10 consecutive digits) appears within the first 200 million digits of Pi.
Search here: The Pi-Search Page
Not always. Take the Celsius and the Kelvin temperature scales. There is no temperature at which °C = K. The reason for this is that the interval between two temperatures 1 °C apart is the same as the interval between two temperatures 1 K apart, but the two scales use different null points. Of course, this is because the Kelvin was deliberately designed that way; the intervals are the same as in the Celsius scale, but the null point of the Kelvin scale has been moved down to the absolute zero, at -273.16 °C, so that the Kelvin value of a given temperature will always be its Celsius value plus 273.16.
But this is not the same interrelationship as exists between the Celsius and the Fahrenheit scales. These two use both different null points and different intervals of one degree, and under such circumstances there will, indeed, by necessity be some temperature at which °C = °F.
Simply there is something going on that we can’t understand.
Jung
Do you think that something is outside of your head?
What Jung really said was that processes inside human brains / mind sometimes connect meanings to things that are actually factually unrelated. The “mystery” is entirely why this bug in human cognition exists and how it operates.
All the rest is ignorant credulous woo.
Nothing to see here. Move along now.
That Wikipedia article says synchronicity is pseudoscience. Which it is. The ramblings of the early psychologists have been abandoned by science for a reason.
Good point.
And the map on the table analogy would also fail if the map is at exactly 1 to 1 scale.
That would be a remarkable coincidence, wouldn’t it?
Cool! And I actually think I know why it works. 
I occupied an apartment in Zurich for a year. Outside the bathroom window was a thermometer that had two scales, Celsius and Reaumur. I happened to notice that if added the two values and added 32 you always got the Fahrenheit temperature. Of course this is obvious once you realize that Reaumur scale is precisely .8 of the Celsius scale. But a surprising coincidence nonetheless.
Typing 314159 (6 digits of a 10 digit #) as a phone number into my client database returns one result. I actually remember this when they first called, and it still makes me grin when I see it pop up on caller ID.
This is out of around 4000 clients.
And Lou Gehrig died of Lou Gehrig’s disease. What are the odds of that???
I suppose the first point is that there are so many potential occurrences, but as the Feynman license plate quote brings to our attention - we only focus on the occurrences that jump out at us as coincidence and ignore the thousands of experiences which do not raise to coincidence.
For example, I might go years without running across a reference to (picking several at random) Oona Chaplin or Nils Bohr or Andersonville, then hit two completely unrelated references to that within days. But, I get megabytes of data spewed at me at random every day. Coincidences happen.
The other point, to pick up on the OP’s discussion of leap years - there is no correlation between two intervals, solar day and solar year - so any attempt to relate the two will be an approximation. 1 in 4 didn’t fit exactly. So 1 in 4 minus 1 in 100 plus 1 in 400 fits better. But millennia from we will have to make a finer adjustment, no matter what number base we choose. The same could be said of moon cycles in a year or days in a moon cycle - there is no direct dependency, so any numeric description will be an approximation with a some error.
Not quite sure where you are going with that. For Brouwer’s Fixed Point Theorem to work the two sets must be that same so if a 1-1 scale map of the US were placed over the US there would be (at least) one spot under it’s spot on the map.
Missed the edit time
If you are saying that a 1-1 scale US map were folded up on a table then it would still work since the point where the desk is is on the map so it is mapped to itself.
In fact, occasional one-second (“Leap Second”) adjustments are made now and then, somewhat irregularly, to prevent the drift from exceeding one second. It turns out that even this causes problems, so the practice is controversial.
If the map is 1:1 scale and you lay it over the mapped area (so every point corresponds), then slide it 1cm to the west, no point will correspond.
I think for the theorem to be guaranteed to work, the map has to be placed so it is fully contained within the boundaries of the mapped area (which will almost invariably mean it must be at a smaller scale than 1:1)
And that is analogous to the example of the Celsius versus Kelvin temperature scales, which are simply apart from each other by a fixed value of 273.16.
The map-on-the-map theorem works like this:
You have a map spread out flat on the table. You have another identical copy of the map sitting on top of the flat copy, but it is folded or crumpled up so that it is entirely contained within the area of the flat map (as Mangetout says above). That is, every point of the crumpled map is directly above some point of the flat map. (I’m not sure if there is any requirement that the maps be at the same scale.)
Then there will be at least one point on the top map that is directly above its counterpart point on the bottom map.
ETA: However:
This fails the required conditions, as it will leave the top map with some points that are not directly above any part of the lower map. The theorem requires that the top map lie entirely above the bottom map.
ETA: I think there is also a requirement that neither map have any gaps or be torn in any way. The two surfaces must be continuous.