isnumeric() with PostgreSQL

You problem is the two 0 or more [0-9] elements on each side of the decimal point. You need to use a logical OR | in the number identification line:

~'^([0-9]+\.?[0-9]*|\.[0-9]+)$'

This will exclude a decimal point alone as a valid number.

As you may noticed, regex-based method is almost impossible to do correctly. For example, your test says that 1.234e-5 is not valid number, when it really is. Also, you missed negative numbers. What if something looks like a number, but when you try to store it it will cause overflow?

Instead, I would recommend to create function that tries to actually cast to NUMERIC (or FLOAT if your task requires it) and returns TRUE or FALSE depending on whether this cast was successful or not.

This code will fully simulate function ISNUMERIC():

CREATE OR REPLACE FUNCTION isnumeric(text) RETURNS BOOLEAN AS $$
DECLARE x NUMERIC;
BEGIN
    x = $1::NUMERIC;
    RETURN TRUE;
EXCEPTION WHEN others THEN
    RETURN FALSE;
END;
$$
STRICT
LANGUAGE plpgsql IMMUTABLE;

Calling this function on your data gets following results:

WITH test(x) AS ( VALUES (''), ('.'), ('.0'), ('0.'), ('0'), ('1'), ('123'),
  ('123.456'), ('abc'), ('1..2'), ('1.2.3.4'), ('1x234'), ('1.234e-5'))
SELECT x, isnumeric(x) FROM test;

    x     | isnumeric
----------+-----------
          | f
 .        | f
 .0       | t
 0.       | t
 0        | t
 1        | t
 123      | t
 123.456  | t
 abc      | f
 1..2     | f
 1.2.3.4  | f
 1x234    | f
 1.234e-5 | t
 (13 rows)

Not only it is more correct and easier to read, it will also work faster if data was actually a number.