Wow - that's really odd! No, there isn't anything in the MLA settings you've missed. It must be a bug or perhaps an interaction with another plugin on your site. I tried downloading and activating the Enable Media Replace plugin, but it seems to work fine and doesn't interfere with MLA on my test system. Do you have other plugins in place?
There are two places to start looking for the cause. First, is the data getting from the server to your browser. If you look through the HTML source for the Media/Assistant submenu page you should be able to find the data for each attachment. It will look something like this:
<div class="hidden" id="inline_7">
<div class="post_title">David Lingren Blue Background Title Keyword</div>
<div class="post_excerpt">David Lingren Blue Background CaptionKeyword</div>
<div class="image_alt">David Lingren Blue Background AltKeyword</div>
<div class="mla_category" id="category_7">5,1</div>
<div class="mla_tags" id="post_tag_7">Test Tag</div>
<div class="mla_category" id="attachment_category_7">33</div>
mla-inline-edit-scripts.min.js in the list. If it's not there, that's a problem.
The other experiment you can try is to make a change or two, click on "Update" and see if the changes make it back to the server and get recorded with the attachment. Try this on a test attachment of some sort so you don't break anything on your site.
You can also try the quick edit feature on a post or page and verify that the normal WordPress feature is working for you.
Finally, can you tell me what browser you're using and whether you have a Mac or Windows system? Since the browser is involved, I want to make sure I'm testing in your environment.
I am sorry you're having this issue and I'd like to resolve it. If you can give me more information it would be most helpful.
Thanks for this report, for your interest in the plugin and your patience.