For starters, you are right, the number is based on the number of countries that are currently either members of the United Nations (UN) or observer states.
There are 193 countries that are members of the UN--see here-- and 2 countries that are observer states, Vatican City and Palestine--see here for the full list.
Now, as far as Socratic is concerned, the list stops at 195 countries because that's how many countries are being recognized by the US Department of State--see here. Since Socratic is an American company, it makes sense to follow the official position of the US on how many countries are recognized as independent states.
So there's really nothing "wrong" with the other 13 countries, it's just that they're not officially recognized as independent states by the US Department of State.
ADDENDUM
It's worth mentioning that the State of Palestine, which is a non-member observer in the UN, is not recognized by the US Department of State as an independent country.
The State of Palestine's place on the list of 195 countries recognized by the US Department of State is taken by Taiwan, which is not a member of the UN but which receives special status from the US, despite the fact that the US officially recognized in 1979 that Taiwan is part of China.
