It turned out that the Irfanview website still lists the 32-bit plugins as default on their website, and it's not clear at all that this won't work for the 64-bit version of IrfanView (it's not clear that there are 64-bit plugins in the first place, but also the site actually says that most 32-bit plugins will work automatically).
Formats.dll is one of the plugins that needs to be downloaded in its x64 version. Their website has a link to 64-bit plugins in the plugin page, so if you are running Irfanview x64, make sure to get that.