It is not cold war. Western and eastern Europe is an historical subdivision that traces back to roman empire. Check how the cold war separation matches previously existing boundaries.

Italy is both south and western Europe.

Edit. as you added the link on UN. I n the same page you link, check

https://en.m.wikipedia.org/wiki/Western_Europe#UN_regional_groups:_Western_European_and_Others_Group

The part you point out (according to the page itself), it just for convenience of statistical reports. For political/historical purposes, Italy is western Europe