My question is about the size breakpoint of Fama French portfolios. Anyone knows why in US market data they used the median as a size breakpoint to construct the six portfolios, but when they used the developed and emerging markets data they used 90%/10% as a size breakpoint?
With respect to size in the U.S., Fama and French calculate breakpoints from the NYSE sample only, but apply the breakpoints to the whole sample of NYSE, AMEX, and NASDAQ stocks. You can find this definition on Kenneth R. French's website here and here.
The authors aim to mirror the NYSE breakpoints for all markets in order to keep it consistent (comparable). Even for non NYSE in the US you do not have the median anymore, but the difference is more pronounced for emerging markets.
Size, value, and momentum in international stock returns by Fama French clarifies this choice too:
[...] 90% of market cap corresponds roughly to the NYSE median, used to define small and big stocks in Fama and French (1993).
On the Construction of Common Size, Value and Momentum Factors in International Stock Markets: A Guide with Applications by Schmidt et al discusses this in section 3.2 on page 12.