Anthem will release its official release on February 22. Then, PS4 owners will also be able to play for the first time. However, some are already doing so now.
Who can play Anthem now? For players who have a physical CD of Anthem, the servers are already online (February 20).
This applies to both PS4 and Xbox One.
Players are reporting this on Reddit and also in the MeinMMO comments.

When can everyone play? On February 22, at 00:01 (midnight) is the official release. Then the servers will also open for digital pre-orderers.
Day-One Patch arrives only with release
What is the Day-One Patch? The first major patch of Anthem fixes many issues and bugs. Here we have 17 highlights from the Day-One Patch. This includes improved load times, the eternal loading screens. Additionally, the Javelins, such as the Interceptor, will be adjusted.
Should I play without the Day-One Patch? If you are playing now with the physical copy, then you also have the same issues and bugs that PC players currently have. If this does not bother you, there is nothing against playing now.
However, if you are really annoyed by bugs that cause crashes, then you should wait for the first major patch.

When does the Day-One Patch come? It will be applied on February 22, when Anthem has its official release.
Do you want to know more about the PS4 release? Then check out our comprehensive overview. Here you will find more backgrounds about Anthem, such as the story and gameplay. We also address technical aspects of the PS4 in connection with Anthem:
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Anthem – MeinMMO