Here a bit longer the way to get there
The first hint excludes all sums equal to 2, 14, 16, or >=18. Otherwise there would be always (at least) one option where Bernard could know the birthdat.
The second hint excludes all dates for which only combinations where product and sum are the same exist, e.g. 5/1 and 1/5.
That reduces everything to the following dates:
product 4: 04/01, 02/02, 01/04
product 9: 09/01, 03/03, 01/09
product 14: 07/02, 02/07, 01/14
product 16: 08/02, 04/04, 02/08, 01/16
product: 36: 12/03, 09/04, 06/06, 04/09, 03/12
product 42: 07/06, 06/07, 03/14
Because Albert still doesn't know, we exclude 02/02, 03/03, 04/04, and 06/06 (unique sums). Bernard could now conclude that the puzzle is insolvable if the product were 4, 9, or 36. As he doesn't do so, we can exclude those dates as well.
Of the remaining dates
product 14: 07/02, 02/07, 01/14
product 16: 08/02, 04/04, 02/08, 01/16
product 42: 07/06, 06/07, 03/14
only 01/14 has a unique sum.
Edit:
I noticed that I forgot the product 30: 10/03, 06/05, 05/06, 03/10, 02/15, but that doesn't change the result.