I would not recommend using MSYS2 to work with python-igraph unless you have a very good reason. We provide pre-compiled Windows binaries for the official Windows Python (which I believe is not binary compatible with the one in MSYS2). You can also get python-igraph for Windows from Anaconda.
If you want to use MSYS2 anyway, you will need to compile python-igraph from source. However, you may not need to compile igraph’s C core from source as well. Instead, install it from MSYS2:
https://packages.msys2.org/package/mingw-w64-x86_64-igraph?repo=mingw64
Now check the pkg-config --cflags --libs igraph again. This time it should work.
If you want to let the C core be compiled from source as well, be sure to check the instructions: igraph Reference Manual You must be consistent with using the mingw64 msys terminal, and installing the mingw-w64-x86_64 variants of tools. This means that you should not install the cmake package, but the mingw-w64-x86_64-cmake one.
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