1. Install the first instance of cygwin in a partition.
2. The installation makes the following registry entries.
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions]
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin]
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin\mounts v2]
"cygdrive prefix"="/cygdrive"
"cygdrive flags"=dword:00000022
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin\mounts v2\/]
"native"="C:\\hide-cygwin"
"flags"=dword:0000000a
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin\mounts v2\/usr/bin]
"native"="C:\\hide-cygwin/bin"
"flags"=dword:0000000a
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin\mounts v2\/usr/lib]
"native"="C:\\hide-cygwin/lib"
"flags"=dword:0000000a
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin\Program Options]
3. Export the registry tree of HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions. If help is needed with exporting registry, refer to this link.
4. Then copy cygwin installation folder to another location (partition), and make a copy of the exported registry file. Open the copied registry file and change the contents to reflect the partition change.
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions]
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin]
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin\mounts v2]
"cygdrive prefix"="/cygdrive"
"cygdrive flags"=dword:00000022
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin\mounts v2\/]
"native"="D:\\cyg2win"
"flags"=dword:0000000a
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin\mounts v2\/usr/bin]
"native"="D:\\cyg2win/bin"
"flags"=dword:0000000a
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin\mounts v2\/usr/lib]
"native"="D:\\cyg2win/lib"
"flags"=dword:0000000a
[HKEY_LOCAL_MACHINE\SOFTWARE\Cygnus Solutions\Cygwin\Program Options]
5. Now I have 2 exported registry files. And I am going to create 2 shortcuts to bash binary file like this, C:\hide-cygwin\bin\bash.exe --login -i. And another shortcut pointing to D:\cyg2win\bin\bash.exe --login -i.
6. Finally, the cygwin instances can be run successfully. When one of them is about to run, the related registry file needs to be applied. Otherwise, bash would have problems.
Disclaimer: I am not saying this is the best solution. At least it solved my problem.