The unix utility join will do this (assuming that file1 and file2 are sorted):

    $ join -t\| file1 file2

    A1|dog||Fido|
    A2|cat||Fluffy|
    A3|bird||Tweety|
[download]

From which you could filter out the extra empty field or extend the command line options to select specific fields:

    $ join -t\| -o 1.1 1.2 2.2 2.3 file1 file2

    A1|dog|Fido|
    A2|cat|Fluffy|
    A3|bird|Tweety|
[download]

--
John.