hmmmmmmm..

https://test.wikipedia.org/wiki/GoodEqualsExample:

{{=}}

works as expected, yet your example:

{{=|arg}}

does not — I have a sneaking suspicion that this is due to variable magic words not supporting parameters (like that). Parser function magic words (which are separate from ParserFunctions because of course they are) do support arguments like that.

Due to this type of magic word not supporting an argument, it appears to be falling back to the still-existing template (which doesn't exactly explain why that category is being added as the template is blank)

(and yes, T91154: {{=}} should be a parser function called for it to implemented as a parser function, but it was added as a variable instead)

Create a page with {{=}} on it, such as https://test.wikipedia.org/w/index.php?title=BadEqualsExample&oldid=517742

To be clear, you're actually doing {{=|arg}}, which isn't supported and is actually invoking a template, like the error indicates.

Here's what you get when you preview that:

Note the "templates used in this preview" includes Template:=. I don't know if this is the ideal behavior, but it does match how {{!}} works, see:

So, despite {{=}} being magic, if passed arguments it bypasses the magic? Is that by design? That is what I'm seeing in the output at https://test2.wikipedia.org/wiki/EqualsTest2 (when Template:= is deleted from that project)

It has been that way for some time. enwiki has Category:Pages which use a template in place of a magic word to track such uses.

OK, if this is expected behavior nothing to do then I suppose.