The idea of using the push notifications is only to have a notification mechanism that will send a notification when there is a new message waiting to download from the backend service: all the notifications are of the same type e.g. "refresh messages from the server" (the same for Android and iOS).
Are there any limits for the Apple/Google push notifications services? Assuming that my application will handle more that 100k active users (or even 1M or more users) - would there be any problem with the Apple Push Notifications or GCM services?
If using the push notifications for such a service is not a good idea then what is other solution that could be used for mobile chat applications?
Don't forget that on iOS user may forbid sending notifications, so your app won't receive any even in foreground. Thus, you need to implement your own push mechanism.