I'm trying to figure out which service should I use to send push notifications for cases very similar as to when an app like Instagram notifies you every time you receive a new like, or as when Tinder let's you know a new person liked you.
I'm kind of confused, I'm not sure if a service like Amazon SNS or Amazon Pinpoint would be a good use case for this.
Maybe it's a completely different service I'm looking for?
You can definitely achieve this using SNS. Using SNS, you can set up topics to target subsets of your customers (e.g. broadcast a notification in German to all of the customers in Germany) as well as send push notifications to individual devices.
Depending on whether you are targeting Apple, Android, web or a combination of all three, tutorials can be found showing precise steps to set this up.