I disagree on removing pgadmin tag / renaming to pgadmin-3 - but not strongly on adding it to old questions that apply.
With only 160 questions tagged pgadmin, I don't see any benefit on renaming.
PgAdmin 4 is a total rewrite.
PgAdmin 4 is a total rewrite, yes. But do we really care if the program is written in C or Python? The functionality is very similar. The writers kept the same name after all. If it was renamed, it might be reasonable to use the new name as a tag.
PgAdmin3 is a legacy product that is doomed to die.
I suggest we wait until that happens. We've been getting questions (rare but still) about Postgres 7, 8.0, SQL Server 2000, MySQL 4.1, all products that are years past their end of life / support. I'd bet we'll get questions for PgAdmin3 for years to come, even if they become increasingly rare. After all, it has been - for 14 years - one of the most popular interfaces for accessing Postgres. Habits don't change easily.
All of the bugs, and interfaces on it are not applicable to PgAdmin4.
The bugs that might appear may of course not be similar but we can always tag with both pgadmin and pgadmin-4 questions regarding the new version - and with both pgadmin and pgadmin-3 for new questions regarding the old version.
No need for retagging the old questions either - although I wouldn't disagree on adding pgadmin-3 tag to the old ones that apply, if that can be done easily and without disrupting the site's front page.