This reproduces for me (M57 stable and ToT build of Chromium).
Looking at the audio player element in developer tools, Chromium has a
<div class="me-cannotplay"> ... </div> element while M57 stable Chrome
only has an <audio> element (See attached screenshots).

I'm not familiar with authoring with wordpress, but it looks like the element
is inserted by this plugin:
http://www.mediaelementjs.com/

And probably the plugin detects browser feature or something and inserts the
"class=me-cannotplay" download link (because the audio player feature is not
detected?).

I'd suspect this is a problem that Chromium lacks some codec to play the
specified audio, or something and this is not a bug in Chromium.

orsch...@, could you confirm?

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kochi,

I don't know which codec it could lack since the elements display correctly on Chrome. Unless, of course, Chrome has these codecs built-in.

Okay, can someone responsible for media control element confirm?
(Adding Internals>Media for wider visibility)

This is not my area so it is still a guess, but I thought 
Chrome has more codecs than Chromium and that was the reason
the player plugin element chose to show "download" link instead
of built-in player element.

You can check which codec is supported by Google Chrome but is not
by Chromium: https://www.chromium.org/audio-video

Yes, Chromium doesn't support as many codecs as Chrome (for licensing reasons, for example).
Do you need support in a Chromium build for testing?
You should be able to add:

media_use_ffmpeg = true
media_use_libvpx = true
proprietary_codecs = true
ffmpeg_branding = "Chrome"

to your args.gn and rebuild - Chromium will have the same codecs as Chrome then.

I am sorry for my lack of technical understanding. 

I am only an average user who likes the Chromium browser.

However, finding popular websites such as WordPress not working properly is a bummer. 

Is there anything the end-user can do that does not involve compiling and rebuilding of Chromium, neither completely switching to Chrome?

Yours,

Robert

Google doesn't distribute pre-built Chromium, only distributes Google Chrome,
so building Chromium is not what an average end-user would do, I suppose.

Linux users are kinda special - because some distributions only bundles
Chromium, not Google Chrome.
Probably there are reasons why some people prefer Chromium to Google Chrome,
one reason may be to avoid proprietary codecs...

So, this is something that needs to be fixed on the WordPress site?

Let me explain further.  I assume you understand some technical terms like
"codec", "compile" etc.

The wordpress site or wordpress itself isn't a real issue.
The site quoted in the original report embeds "MediaElement.js" plugin
for embedding the player in the blog content.
But this is not a real issue, either. So no, the site do not have to be fixed.

But yes in a sense, that one way to solve the issue is to upload the audio content
encoded in some codec that Chromium has (e.g. Ogg Vorbis).
Other way to solve this issue is to use Google Chrome.

The audio content on the page seems to be in "mp3" format, which requires one of
the codecs that Chromium lacks.
The reason Chromium lacks some of the codecs available in Chrome is licensing etc.

If you are using Chromium on x86-64 Linux, probably you can run Google Chrome
on that machine without much effort.  So I was curious why you persist on
Chromium, not Google Chrome for this matter.

From the comments from you so far, I assumed you are using Chromium that is
bundled with your distribution or via some package that is available to the
distribution, and you don't compile Chromium yourself.
If you compile Chromium yourself, the comment4 is the way to fix this issue.

So technically Chromium can include codecs that is not enabled by default,
but for some reason the packager for your distribution chose not to enable them.

Some distributions seem to provide a package for codecs not enabled in Chromium
by default, e.g.
http://packages.ubuntu.com/search?keywords=chromium-codecs-ffmpeg-extra
so installing them might solve your issue (I haven't tested).

Dear Kochi, 

I appreciate all your help!

The mentioning of codecs was very helpful for me to understand the situation better. 

Unfortunately, however, I have already installed the latest version of chromium-codecs-ffmepg-extra (56.0.2924.76-0ubuntu0.16.10.1335).

The fact that it is not working tells me that there must be more Ubuntu users affected to play .mp3 files inside Chromium, correct?

I might need to get the feedback of more Ubuntu users on this...

Yours,

Robert

Hmm, I'm sorry to hear installing the codecs still didn't help.

I think it is an issue of Ubuntu, so this issue probably is better discussed
in Ubuntu's user forum and there would be more probability of getting useful help.
I cannot help more about how Ubuntu provides codecs etc.

Thanks,

Dear Kochi, 

Absolutely no problem. 

I really appreciate your help so far. 

I have asked the question on AskUbuntu [1] and hope someone there can help me. 

Yours,

Robert

[1] https://askubuntu.com/questions/897164/what-packages-are-required-to-play-mp3-files-in-chromium

Just FYI, following the recent expiration of mp3 patents, we might include
mp3 codecs in Chromium (note: not Google Chrome!) builds.

Follow the issue for the progress:
https://bugs.chromium.org/p/chromium/issues/detail?id=746579

Note that even we enable mp3 codecs in non-proprietary builds, some
distribution can ship Chromium without mp3 codec integrated. YMMV.