I like Postcrossing for one of the same reasons I like the Olympics: It’s an opportunity to bring the world together. I decided to pull some statistics to see if there was something interesting when you compare a nation’s participation in Postcrossing and its participation in the Olympics.
My methods:
 I used the data supplied on Postcrossing to get a list of nations, their populations, number of members, and number of postcards sent.
 I used the data supplied on Wikipedia to get a list of number of athletes participating in each nation during the 2020 Olympics.
 Because the list of nations for the Olympics is not the same as the ones used for Postcrossing, I had to modify the statistics slightly. (North Korea and Vatican City are not participating in the Olympics, and territories like Greenland, Norfolk Island, etc. that Postcrossing treats separately, I added to the sovereign nation that claims it.)
 I then performed some calculations to note trends.
You can see the results on the spreadsheet here.
How to interpret the data:

Countries highlighted in yellow are in the Top 10 of either Population, Number of Postcards Sent, Number of Postcrossers, or Number of Athletes.

Columns:
 Country Name
 Population
 Number of PX (Postcrossing) Members
 Number of Postcards (PC) Sent
 Number of Athletes at the Olympics
 Number of Postcrossers per 1,000,000 people
 Number of Postcards Sent per 1,000,000 people
 Number of Olympic Athletes per 1,000,000 people
 Ratio of Postcrossers to Athletes
 Ratio of Postcards Sent to Athletes
 Ranking of the previous 9 columns
 The difference between the rankings of Postcrossers and Athletes
 The difference between the rankings of Postcards and Athletes

For the ratios of Postcrossers or Postcards to Athletes, the higher the number, a number greater than 1 indicates that there are more postcrossers or postcards than there are athletes. The higher the number, the greater that ratio is.

Also for the ratios, if the number is exactly 1, then there are the same number of postcrossers or postcards to athletes.

If that number is between zero and 1, then there are more athletes than there are postcrossers/postcards.

If the number is zero, then there are no postcards/postcrossers.

For the last two columns (differences), if the number is close to zero, then the ranking of the number of postcards/postcrossers ranks about the same as the ranking of the number of athletes. (That is, the popularity of Postcrossing is about the same as the Olympic participation.)

If the difference is a larger negative number, then Postcrossing is comparatively more popular.

If it is a larger positive number, then Olympic participation is comparatively more popular.
Some interesting observations that I’ve seen so far:
 Brunei is the place where Postcrossing is most popular, compared to the number of Olympic athletes. There are over 900 postcards sent and over 17 postcrossers per 100,000 people there, but less than 0.5 athletes per 100,000 people.
 The differences in the ranking between Postcrossing popularity (both Postcrossing members and postcards sent) and Olympic participation is 5 or less for US, China, and Germany. (US is #1 in Olympic athletes, #3 in Postcards, and #4 in Postcrossers; China is #5 in Athletes, #6 in Postcards, and #3 in Postcrossers: Germany. is #4 in Athletes, #1 in Postcards, and #5 in Postcrossers). This is to say that the US, China, and Germany are just as active in both Postcrossing and the Olympics. (You didn’t need data to back that up, but now you do.)
 Similar balanced rankings for nonTop 10 countries include: Cayman Islands, Turkmenistan, Puerto Rico, Panama, Croatia, Qatar, and Palestine.
 There are just as many Postcrossers as Athletes in Dominica, BVI, Benin, Cape Verde, Monaco, and Antigua & Barbuda.
I’m sure my methods aren’t totally accurate, and there are plenty more observations that can be made. Feel free to point out anything else you find interesting!