There are numerous bugs out there, which may cause Putty not to show the so-called “jump list” (these are the “recent” items on a pinned icon on the task bar) under Windows 10. Amongst them are:

• https://superuser.com/q/1009597
• https://www.chiark.greenend.org.uk/~sgtatham/putty/wishlist/win-jumplist-trouble.html

However, you should also check that this feature is enabled in Windows 10 in general (which wasn’t the case with mine). Details on that you may find at https://www.raymond.cc/blog/clear-and-disable-jump-lists-in-windows-7/