Xcode-select -p returned /Applications/Xcode.app/Contents/Developer, and clang -version reported an installation directory under there. xcode-select -r and rebooting didn't solve this issue. But clang -version still gave 8.0.0 as the version number.
The App Store didn't suggest this upgrade, and neither did softwareupdate -list. I wanted to upgrade the command line tools from version 8 to 9.
Hopefully it'll help someone (and it'll surely help me next time I run into the same issue). I'm going to answer a slightly different question here, because this question came up when I searched for a solution to my problem.