Well that depends on how harsh you want to scrutinize the story. I do not think the Khan Academy story is wrong in itself, although it lacks nuance for sure. The story is oversimplification of a reality, but I assume their macro course is targeting high school kids, not undergraduates or graduates. A high school treatment of a topic won't have the same level of nuance as graduate treatment. For example, a high school historical map might oversimplify political situation by ignoring nuances such as this or that area might be assigned the same color as a rest of a country even though it was historically perhaps disputed area.
Note that the story is carefully written (see my emphasis):
One morning, everyone in the Land of Funny Money awakened to find that the monetary value of everything had increased by 20% (...). Every price in every store was 20% higher. Paychecks were 20% higher. Interest rates were 20% higher. The amount of money—everywhere from wallets to savings accounts—was 20% larger. This overnight inflation of prices made newspaper headlines everywhere in the Land of Funny Money. But the headlines quickly disappeared as people realized that, in terms of what they could actually buy with their incomes, this inflation had no economic impact. Everyone’s pay could still buy exactly the same set of goods as it did before.
The way as I read the story is to illustrate that the nominal variables don't really matter as opposed to real variables. The story is also correct as you yourself notice because it assumes the amount of money in wallets increased as well. In fact this story that Khan Academy says is a retelling of Hume's (1752) famous thought experiment from Political Discourses about doubling England's money supply overnight.
In the anecdote, the money stored in cash and savings accounts also grows by 20%, so indeed people in the Land of Funny Money are as well off as before. But that doesn't happen in real life, right?
Not generally, although that depends on what caused the inflation. It is generally accepted by economists that inflation is caused (among other things) by expansion of money supply (See Mankiw Macroeconomics). Government via central bank could decide to send every living person some sum of money like in the example from Khan Academy. For example, in the US Covid19 stimulus was largely funded by newly created money since a lot of US debt was bought by Fed (see this brookings article). It is not the same as the story above as the size of stimulus check was fixed, but it is somewhat similar.
However, you are right that generally speaking inflation will not increase value of your saving that you held in forms of cash or cash deposit. This will create some distortions in the economy for sure.
Nonetheless, again the way how I read the Khan Academy story is that it is a simple high-school level explanation explaining that inflation is not as horrible as a non-economist might think and also showing the important distinction between nominal and real variables. For better or worse truth is that any high school textbook could be completely destroyed by PhD level experts in a given field if the intention would be to scrutinize the text in a harshest way possible. There is a trade-off between offering simple explanation and 100% accurate one.
Yes, you can shield your wealth from a 20% inflation by investing it on 20% interest rate bonds. But if there was no inflation, you could get 20% richer by buying those bonds! Isn't inflation really making you poorer in that case?
No this is not correct. The Khan Academy does not make it clear, but they are obviously (to an economist) talking about nominal interest rate. Nominal interest rate is given by the Fisher equation as:
$$ i \approx \pi + r$$
where $i$ is the nominal interest rate, $\pi$ is inflation, $r$ is the real interest rate.
The real interest rate reflect how much people demand to be compensated for investing/lending money regardless of the inflation rate. The real rate of interest is also what you actually in real terms get for investing/saving. In the Khan Academy story they implicitly assume real interest rate is 0. So in that case if there would be zero inflation you would also get 0 interest rate, if there is 2% inflation you get 2% interest rate.
Point is inflation does not cheat you on your interest rates unless you lent money using some security that pays fixed interest rate. For example, if you are a bank and you give someone 30-year mortgage at fixed 5% interest rate (when inflation was zero), and then inflation increases to 10% you are loosing 5% in real terms. However, even in this case bank will A) try to anticipate future interest rate changes B) charges you higher interest rate because of the risk that interest rate changes (this is why variable interest rate mortgages, ceteris paribus, offer lower interest rates than mortgage with same terms but fixed interest rate).
