There was no such issue in the previous versions of Live Messenger. It would automatically log you into Hotmail regardless of your default browser.
Jonathankay has created a little tool called Hotmail in IE to solve this problem. This tool will allow you to continue to use your default browser in Windows, but use Internet Explorer for Messenger’s Hotmail features.
How to use: Just download the executable and run it (no installation required). You will be get a window as shown below:
Choose the Internet Explorer button and select ‘Done’ that’s all, no further changes should be needed.
Supports Windows XP, Windows Vista and Windows 7(Both 32-bit or 64-bit).