No, it's invoking the target g++ compiler and that is picking up the host includes. If I manually run what that makefile is doing:
and looking at line 10 of that file, it's just:
So the include path for the g++ in the arm-none-eabi-gcc port appears to be wrong.
% arm-none-eabi-g++ -c /home/arg/work/pico/pico-sdk/src/rp2_common/pico_standard_link/new_delete.cpp In file included from /usr/include/c++/v1/cstdlib:84, from /home/arg/work/pico/pico-sdk/src/rp2_common/pico_standard_link/new_delete.cpp:10: /usr/include/c++/v1/__config:1132:6: error: #error "No thread API" # error "No thread API"