Are all numbers in (20;30] 50th percentiles and medians of set of numbers {10,20,30,40}? (Here I implicitly use the exclusive definition of a percentile)
Let’s assume that we have following set of numbers: {10,20,30,40}.
The median of this set is the mean of 20 and 30, namely 25.
But here is a problem: Our median is basically 50th percentile. And 50th percentile means that 50% of datapoints are below our number, namely below 25. And it’s true! BUT, while it’s true for 25, it’s also true for any number in (20;30] interval, like 23. Does it mean that all numbers in (20;30] interval are 50th percentiles and consequently, medians?  And if not - why?
 A: Yes, you are correct. The median of a distribution is not always unique.
A: Your observation is correct. That said, this ambiguity comes up much more when you learn about medians with small examples than when you see medians used for real data. There is no ambiguity when you see a statement like 

The U.S. Census Bureau reported in September 2017 that real median
  household income was $59,039 in 2016, exceeding any previous year.(*)

In fact, reporting "about \$59,000" would be better than the meaningless precision of the extra \$39.
You sometimes see a similar problem when asking for the mode, if there are exactly two duplicate values. That statistic is useful for large data sets partitioned into bins, for a histogram. Then the mode is the range for the nighest bar.
(*) https://en.wikipedia.org/wiki/Household_income_in_the_United_States
A: 
Our median is basically 50th percentile.

Incorrect. The median and a 50th percentile are NOT the same thing. There is only one method for finding the median, while there are many methods (up to 8, probably even more. Calculator for all 8 methods: https://www.wessa.net/rwasp_percentiles.wasp) to find a percentile and they can give such 50th percentile that it can be different from the median. The median gives similiar, or even equal result, to 50th percentile, but it does NOT mean the same thing, it does NOT imply that "50% of datapoints are below the median", although it CAN be used as estimator for 50th percentile. 
So there can be several 50th percentiles, but the median is always only one.
