Google employee? See go/chrome-gms for more info.
The Google Play services are a combination of services exposed on Android devices and libraries to interact with them. Chrome relies on them for critical features like Sign in, Feedback or Cast.
The standard way of adding the Google Play services as a dependency to a project is to import it through the Android SDK manager as a Maven repository. That repository contains multiple versions of the library split into separate APIs (for example Cast, GCM, Android Pay, etc). To avoid downloading a lot of data we don't need to build Chrome, android checkouts of Chromium download an archive containing only the APIs we currently need in Chrome, and in a single version.
The up to date list of clients and version used can be seen in //tools/android/roll/android_deps/build.gradle.
To update play services, change the gmsVersion variable in //tools/android/roll/android_deps/build.gradle, then run [//tools/android/roll/android_deps/fetch_all.py][fetch_all.py].
As explained above, the default checkout has access to only a specific set of APIs during builds. If your CL depends on some APIs that are not included in the build, you will need to add an entry in //tools/android/roll/android_deps/build.gradle.
Not doing so could make the CL fail on the trybots and commit queue. Even if it passes, it might fail on the internal bots and result in the CL getting reverted, so please make sure the APIs are available to the bots before submitting.