Is there something like a zip() function in PostgreSQL that combines two arrays?

Postgres 9.5 or later

has array_agg(array expression):

array_agg ( anyarray ) → anyarray

Concatenates all the input arrays into an array of one higher
dimension. (The inputs must all have the same dimensionality, and
cannot be empty or null.)

This is a drop-in replacement for my custom aggregate function array_agg_mult() demonstrated below. It’s implemented in C and considerably faster. Use it.

Postgres 9.4

Use the ROWS FROM construct or the updated unnest() which takes multiple arrays to unnest in parallel. Each can have a different length. You get (per documentation):

[…] the number of result rows in this case is that of the largest function
result, with smaller results padded with null values to match.

Use this cleaner and simpler variant:

SELECT ARRAY[a,b] AS ab
FROM   unnest('{a,b,c}'::text[] 
            , '{d,e,f}'::text[]) x(a,b);

Postgres 9.3 or older

Simple zip()

Consider the following demo for Postgres 9.3 or earlier:

SELECT ARRAY[a,b] AS ab
FROM  (
   SELECT unnest('{a,b,c}'::text[]) AS a
        , unnest('{d,e,f}'::text[]) AS b
    ) x;

Result:

  ab
-------
 {a,d}
 {b,e}
 {c,f}

Note that both arrays must have the same number of elements to unnest in parallel, or you get a cross join instead.

You can wrap this into a function, if you want to:

CREATE OR REPLACE FUNCTION zip(anyarray, anyarray)
  RETURNS SETOF anyarray LANGUAGE SQL AS
$func$
SELECT ARRAY[a,b] FROM (SELECT unnest($1) AS a, unnest($2) AS b) x;
$func$;

Call:

SELECT zip('{a,b,c}'::text[],'{d,e,f}'::text[]);

Same result.

zip() to multi-dimensional array:

Now, if you want to aggregate that new set of arrays into one 2-dimenstional array, it gets more complicated.

SELECT ARRAY (SELECT ...)

or:

SELECT array_agg(ARRAY[a,b]) AS ab
FROM  (
   SELECT unnest('{a,b,c}'::text[]) AS a
         ,unnest('{d,e,f}'::text[]) AS b
    ) x

or:

SELECT array_agg(ARRAY[ARRAY[a,b]]) AS ab
FROM  ...

will all result in the same error message (tested with pg 9.1.5):

ERROR: could not find array type for data type text[]

But there is a way around this, as we worked out under this closely related question.
Create a custom aggregate function:

CREATE AGGREGATE array_agg_mult (anyarray) (
   SFUNC    = array_cat
 , STYPE    = anyarray
 , INITCOND = '{}'
);

And use it like this:

SELECT array_agg_mult(ARRAY[ARRAY[a,b]]) AS ab
FROM  (
   SELECT unnest('{a,b,c}'::text[]) AS a
        , unnest('{d,e,f}'::text[]) AS b
    ) x

Result:

{{a,d},{b,e},{c,f}}

Note the additional ARRAY[] layer! Without it and just:

SELECT array_agg_mult(ARRAY[a,b]) AS ab
FROM ...

You get:

{a,d,b,e,c,f}

Which may be useful for other purposes.

Roll another function:

CREATE OR REPLACE FUNCTION zip2(anyarray, anyarray)
  RETURNS SETOF anyarray LANGUAGE SQL AS
$func$
SELECT array_agg_mult(ARRAY[ARRAY[a,b]])
FROM (SELECT unnest($1) AS a, unnest($2) AS b) x;
$func$;

Call:

SELECT zip2('{a,b,c}'::text[],'{d,e,f}'::text[]); -- or any other array type

Result:

{{a,d},{b,e},{c,f}}