So now, I want to install windows 7. I know that product keys from windows (7, 8 & 10) are used interchangeably.
This is absolutely NOT the case. Windows 7 can only be activated with a Windows 7 license key. Windows 8 can only be activated with a Windows 8 license key. Windows 8.1 can only be activated with either a Windows 8 or Windows 8.1 license key. Windows 10 until December 31, 2017, could be activated with an eligible (Windows 7, Windows 8.x) license key.
Since my Microsoft account is linked to my product key, I can install windows 10, but to install windows 7, I need the key itself.
You will have to purchase a Windows 7 license to accomplish your goal. Windows 7 CANNOT be activated with your Windows 8 OEM license.
Is there any way to see the product key in my MS account?
Windows 7 keys cannot be linked to a Microsoft Account. Windows 8.x keys cannot be used to activate Windows 7.
How can I recover my key?
It is not necessary to recover your Windows 8 license key. The Windows 10 and Windows 8.1 installation environment will automatically detect the license key. While tools exist to display this information, they are not required at any point, to activated Windows 8.x.
Since Windows 8.x license keys CANNOT be used to activate Windows 7, knowing what your Windows 8.x license key is, will not result in your ability to activate Windows 7. You need to purchase a Windows 7 license key to accomplish your goal.