Unterminated dollar-quoted string creating PostgreSQL function

The issue is not with the $$ quoting:

create or replace function test() 
returns void 
as $$
begin
  prepare plan as select 1;
  execute plan;
end;
$$ language plpgsql;
CREATE FUNCTION

 select test();
ERROR:  column "plan" does not exist
LINE 1: SELECT plan
               ^
QUERY:  SELECT plan
CONTEXT:  PL/pgSQL function test() line 4 at EXECUTE

When you run this in the dbfiddle the full error output is:

 Schema Error: error: unterminated dollar-quoted string at or near "$$ begin prepare plan as select 1;"
Schema Error: error: prepared statement "plan" does not exist
Schema Error: error: unterminated dollar-quoted string at or near "$$ language plpgsql;"
Query Error: error: function test() does not exist 

The issue is that EXECUTE inside plpgsql is its own command:

https://www.postgresql.org/docs/12/plpgsql-statements.html#PLPGSQL-STATEMENTS-EXECUTING-DYN

EXECUTE command-string [ INTO [STRICT] target ] [ USING expression [, ... ] ];

I would use the plpgsql form. This works:

create or replace function test() 
returns void 
as $$
begin
  prepare plan as select 1;
  EXECUTE 'execute plan';
  RAISE NOTICE 'Made it';
  DEALLOCATE plan;
end;
$$ language plpgsql;

select test();
NOTICE:  Made it
 test 
------
 
(1 row)