Date and place of birth are commonly used for this purpose in other contexts (although even that is not always enough) but it comes down to what the airline in question required.
Very often, nothing else than the name and credit card number are required (and can therefore ultimately be matched with the booking). Even the nationality is not always required. But in some cases, the airline would have more information (it would depend on the airline but possibly also on the destination, in relation with APIS).
Personally, I don't recall ever providing my place of birth or passport number to an airline at booking time but I never used a round-the-world ticket.
No matter how they enforce it, I expect that all airlines have conditions of carriage that say something like “Tickets are not transferable” or “We will only carry you if you are the passenger named on the ticket”, i.e. they refer to a person, not to a name. Consequently, even if you share so many details with the original passenger that it's not possible to distinguish both of you in practice, you would technically be violating this aspect of the contract by using a ticket that was not intended for you in the first place (you, in fact, don't have a contract and never had one with the airline).