IOS app with real-time updates from server: Socket (using streams) or Apple Push Notification service

Not sure if you got the answer yet. But this is what the documentation mentions

Providers can specify custom payload values outside the Apple-reserved aps namespace. Custom values must use the JSON structured and primitive types: dictionary (object), array, string, number, and Boolean.

So in order to add custom values to your payload, just add them as key-value pairs in your payload. Something like this

{
    "aps":{
        "alert":"Your Message",
        "sound":"push1.wav"
     },
     "custom_key1":"value1",
     "custom_key2":"value2"
}

Here custom_key1 and custom_key2 are your custom keys and value1 and value2 are their values.

Every server has limited capacity to perform certain task. You cant assume that your dedicated server with some xyz configuration will achieve every NFR for you application. You have to do server capacity planning to achieve your application NFR. If your dedicated server is able to send 10K of push notification with in 1 seconds with optimize configuration of server and application, and you want to achieve 20K of notification in 1 second, you have to do horizontal/vertical scaling of server.

In horizontal scaling you put another parallel instance and divide the task between them, however in vertical you scale the same server to have higher configuration. Vertical scaling is somehow not recommended as it has single point of failure.

For your problem to scale you have to put another instance in parallel in divide the push notification sending task between them. For example if you want to send 20K push notification with in a second for 20K devices, you can divide first 10K to first server and next 10K to another one. This will achieve your NFR of sending 20K push notification with in a second.