How to replace multiple special characters in Postgres 9.5

No, there are no this function. Probably is not to hard to write optimized C extension what does it. But C language is not necessary always. You can try SQL or PLpgSQL function:

CREATE OR REPLACE FUNCTION xx(text, text[], text[])
RETURNS text AS $$
   SELECT string_agg(coalesce($3[array_position($2, c)],c),'')
      FROM regexp_split_to_table($1,'') g(c)
$$ LANGUAGE sql;

postgres=# select xx('Jürgen', ARRAY['ä','ü'], ARRAY['ae','ue']);
┌─────────┐
│   xx    │
╞═════════╡
│ Juergen │
└─────────┘
(1 row)

On my comp it does 6000 transformation under 200ms (but I have developer build of PostgreSQL - it is slower).

replace()

If you want just to replace one or few characters you can use function replace(string text, from text, to text) that replaces all occurrences in string substring. The replace function can be used to replace one character to several characters.

translate()

If you want to translate some letters to other letters you can user function translate(string text, from text, to text) that replaces any character in a string that matches a character in the from by the corresponding character in the to set.

Some data to play with:

drop table if exists xyz;

create table xyz (
    id serial not null,
    name varchar(30)
);

insert into xyz (name) values
    ('Juhänäo'),
    ('Jürgüen'),
    ('Dannäu'),
    ('Übüdyr');

Example of replace function:

select replace(name, 'ä', 'a') from xyz;

This function replaces letter ä in the name column with letter a. Juhänäo becomes Juhanao.

select replace(name, 'ä', 'ae') from xyz;

Now it replaces letter ä with ae.

select replace(replace(replace(name, 'ä', 'ae'), 'ü', 'ue'), 'Ü', 'Ue') from xyz;

Not very nice, but in the example all ä become ae, ü become ue, and Ü become 'Ue'.

update xyz set name = replace(replace(replace(name, 'ä', 'ae'), 'ü', 'ue'), 'Ü', 'Ue');

Changes letters and updates rows. The result of the update is following:

Juhaenaeo
Juergueen
Dannaeu
Uebuedyr

Example of translate function:

select translate(name, 'ä,ü,Ü', 'a,u,U') from xyz;

Translates all letters ä to a, ü to u and Ü to U.

update xyz set name = translate(name, 'ä,ü,Ü', 'a,u,U');

Updates table so all predefined letters are translated and the change is saved to the database. The result of the update is following:

Juhanao
Jurguen
Dannau
Ubudyr

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More information:

Replace characters with multi-character strings

Postgresql string functions