Adobe Font List is not a font and therefore cannot be installed?
Are any fonts missing when you open InDesign?
Fonts are not installed via InDesign, they are installed on your computer with any other font.
The dialog box you display is from Windows' Font Control Panel, not InDesign. I assume that you tried to drag-and-drop a font into the Control Panel? Or right click on a font file and use the “Install” option?
The message that your screen shot shows implies that you are trying to “install” a file that isn't a font at all, but rather, a font file list used by InDesign's font rendering system to cache font information. That file is not installable as a font.
What you need to get is the actual font file itself. If it is a TrueType font, it has the suffix .ttf, if an OpenType CFF font, the suffix .otf. If you are getting a Type 1 font from a Windows system, you need both the .pfm and .pfb file associated with the font. If the fonts are from a Macintosh system, only OpenType CFF and Windows format TrueType fonts are cross-platform compatible.
PS: Of course we don't need to tell you that in general, unless the font(s) in question are truly free fonts, fonts that your client has the legal right and license to send to you for use on their projects, or fonts that your organization has its own license for, such e-mailing and installation of such font files is not legal!