Disclosure: I don't use Tinder. This answer is based on general knowledge about Play Services.
What is the reason for this happening?
Looks like Google Play services is not installed on the device or updated. To install it, please refer to Install Google Play Services without Google Play Store. To update it, please refer to Google Play Store or download manually (e.g. from APK Mirror)
Why would Tinder need an active Gmail account on the phone?
As mentioned in the comment by Dan Hulme, "Needing Google Play Services installed isn't the same thing as needing a Gmail account, nor even a Google account." which is partially true regarding Google account (read the next section).
In other words, what Tinder functionality could possibly break?
For Tinder-specific functionality, possibly nothing. However, as already stated in the question,
Failed to Enable Push Notifications. Your Google account credentials are invalid.
The Tinder app may not be able to receive any push notifications (through Google Cloud Messaging). If it is not a concern, then this message can be ignored. However, setting up a Google account is needed on Android 2.x. From Google Cloud Messaging documentation,
It uses an existing connection for Google services. For pre-3.0 devices, this requires users to set up their Google account on their mobile devices. A Google account is not a requirement on devices running Android 4.0.4 or higher.
Could this explain why I am not getting any matches?
As explained in OP's self-answer, no, the matchmaking system has nothing to do with Google Play services and Google Cloud Messaging.