But this standard - EME - isn't for CDMs, it's for the interface between CDMs and Javascript. So yes, your Linux-based browser might well have EME support. But if the company who makes the DRM CDM doesn't support Linux, you're out of luck.
The existence of the EME standard does not in any way increase the likelihood of DRM vendors supporting any more platforms than they do now.
It does make it easier to add support to new platforms, though. If Linux support is just a recompile with a different compiler away (supposing the browser/plugin interface is a simple C API with no GUI), it's more likely to happen.
Please, have a read of the proposal. You'll see that a browser implementing EME does not make it any easier to compile a DRM plugin on a different operating system. The proposed standard has nothing to do with that whatsoever.
All the EME spec specifies is a way to interact with the plugin using Javascript.
The point of HTML5 ECE is to tie into OS-level and hardware-level DRM facilities, so in practice it's going to be far harder to port than existing solutions.
In fact we can see this happening already. Netflix supports multiple DRM schemes, one of which is based on the draft HTML5 ECE standard and is used on ChromeOS. Apparently, both the Silverlight-based player and the Android-based player can be used to watch Netflix on ordinary desktop Linux. The ChromeOS one, on the other hand, only runs on authorised Google-provided hardware and only if you don't enable developer mode; no-one's managed to bypass this yet.
The existence of the EME standard does not in any way increase the likelihood of DRM vendors supporting any more platforms than they do now.