These numbers are way for the company to identify the transaction.
It's never a bad idea to store them. If you do have to speak with someone again it's often pretty convenient to just give them this number, it allows the company rep to pull up the information fairly quickly.
Most of my confirmation numbers are emailed to me so I just hit archive instead of delete.
I would also like to add if you do lose the number, don't worry it's not the end of the world because most companies have another way of identifying you, it just might take a little longer.
Does it provide you any help to identify the transaction?
It might, for instance if I pay Chase and they email me a confirmation(which they do) I can search my email using "chase" or even auto-filter these emails into a label(gmail)/folder(if not gmail). You can in a way do this with the confirmation number, it would be more difficult since you would have to write these down and then search that number.
Is a company going to believe your confirmation number if they can't find it using an internal search?
They wont tell you your a lair, but no your confirmation number won't assist them in finding your transaction. They will probably try to find it some other way instead.