I always assumed the document itself was the (unnamed) root. Makes more sense to me, but apparently that is not the case. Thanks!

I'm actually considering to switch to an entirely different format in future releases. If not, I'll make sure to reorganize and validate the files.

Note the last bullet point in "key syntax rules"https://en.wikipedia.org/wiki/Well-formed_document

There is a single "root" element that contains all the other elements.

Also

When I tried to open anime.xml with Notepad++ (with the XML tools plugin), I got the following popup:

---------------------------
XML Tools plugin
---------------------------
XML Parsing error at line 5: 

Extra content at the end of the document

---------------------------
OK   
---------------------------

One advantage of using XML is that it has very strict rules when it comes to parsing. This makes it easy to detect errors.

I see you're using pugixml to parse XML files. I see pugixml accepts an input like this:

<meta>
	<version>1.3.1</version>
</meta>
<database>
</database>

without returning an error. This means pugixml is a non-conforming parser, because

An XML processor that encounters a violation of the well-formedness rules is required to report such errors and to cease normal processing.

(from the same Wikipedia page; emphasis mine).

So pugixml is not an XML parser. It parses something which is almost, but not quite, entirely unlike XML. I filed zeux/pugixml#337

Luckily, changing the format of history.xml and anime.xml should not be an onerous task (says the guy who doesn't have to do it). If the <meta> element can only contain the version information, that could be moved into an attribute of the <history> or <library> element:

<history version="1.3.1">
etc.
</history>

I'd like to go for this one. The one topic I'm not very sure about is compatibility.
My idea would be simple: Change from

<?xml version="1.0"?>
<meta>
  <version>1.3.1</version>
</meta>
<library>
...

to

<?xml version="1.0"?>
<library version="1.3.1">
...

Should it maybe be called 'taiga-version' or something? (We have an XML version attribute right above it)

And about compat: There's some places like

auto version = ReadXmlVersion(document);
HandleCompatibility(version);

If I change the code to use attributes instead of meta elements, the version that is currently inside the XML files will be ignored (version = 0.0.0). Should the old Read procedure be left in the code for compatibility? Or does this not matter, since there is nothing (except for the version itself) that changed in the XML?

Ok, after reading through the compat code it seems pretty clear.
anime.xml and history.xml do not have downwards-compatibility code since 1.4.0-beta1 (while 1.3.1 still has some code for XML files from versions <1.3.0). This means that even if the version is set to 0.0.0, nothing happens (since there is no compat code).

Now, there is only settings.xml left. The file currently looks like this:

<?xml version="1.0"?>
<settings>
  <meta version="1.3.1" />
  <account>
...

This is different to the other meta elements, and does not violate the 'one root element' rule in its current state. It would be cleaner to change this to

<?xml version="1.0"?>
<settings version="1.3.1">
  <account>
...

BUT there is still compat code for settings.xml. Keeping compatibility for both 'styles' of settings.xml shouldn't be hard, but this change isn't really necessary (as it is already valid XML, even if it is not uniform to the other files). I'd argue for keeping it as-is