Please see
https://en.wikipedia.org/wiki/User_talk:83d40m#PD-FLGov_indentation
The insertion of this tag on a talk page evidently increases the indentation level by approximately the equivalent of four leading colons. The additional indentation is not affected by Template:Outdent or section headings (==...==).
By "tag" you mean inserting the {{PD-FLGov}} into a talk page?
Do you know if this a problem of the code in https://en.wikipedia.org/wiki/Template:PD-FLGov , or a problem of the parser in the MediaWiki code base?
Yes, that's what I mean. You can see the effect twice on 83d40m's talk page:
First in § "File permission problem with File:Map Seagate campus 83d40m University of South Florida Sarasota-Manatee.jpg"
then on the section on capitalization in the article on Antonie van Leeuwenhoek, when I deliberately tested the effect. The user whose talk page this is, 83d40m, thought that I had caused the problem … Well, those details are irrelevant; you can find them in that last-named section.
I don't know where the problem stems from. I don't even know how to get to see the template code, let alone understand how it works.
The template code is here: https://en.wikipedia.org/w/index.php?title=Template:PD-FLGov&action=edit
Looking at the wikitext of https://en.wikipedia.org/wiki/User_talk:83d40m#File_permission_problem_with_File:Map_Seagate_campus_83d40m_University_of_South_Florida_Sarasota-Manatee.jpg you can see
:The image already bears the tag that has been in place since the previous debate.
::::{{PD-FLGov}}Same for https://en.wikipedia.org/wiki/User_talk:83d40m#Casing_of_.22van_Leeuwenhoek.22
As the template is centered on non-user pages I could imagine this to be some feature.