Insert the same fixed value into multiple rows

You're looking for UPDATE not insert.

UPDATE mytable
SET    table_column = 'test';

UPDATE will change the values of existing rows (and can include a WHERE to make it only affect specific rows), whereas INSERT is adding a new row (which makes it look like it changed only the last row, but in effect is adding a new row with that value).

This is because in relational database terminology, what you want to do is not called "inserting", but "UPDATING" - you are updating an existing row's field from one value (NULL in your case) to "test"

UPDATE your_table SET table_column = "test" 
WHERE table_column = NULL 

You don't need the second line if you want to update 100% of rows.

To update the content of existing rows use the UPDATE statement:

UPDATE table_name SET table_column = 'test';