No, unfortunately not. Different countries are using different frequencies for their mobile networks and some networks are using different technologies (CDMA/CDMA2000).
But I think only a few people need worldwide working phones as most European persons never travel out of Europe.
Look at willmyphonework.net if your phone supports the frequencies in the country.
The Fairphone 2 will probably work everywhere, but in some countries only with one, not with two SIM cards (in countries without 2G: South Korea, Japan, soon Australia) and in some countries it only works with 2G/3G, not with LTE (like the USA).
But there are also some China phones which do not support the 800 MHz LTE band, so LTE does not work properly in Europe.
In Europe, the Fairphone 2 (and any other modern European phone) probably works everywhere. It is built for whole Europe, and European networks do not differ much from each other.
For the USA and the Fairphone 2, you should use the T-Mobile network as it’s the only network with GSM, so you can use two SIM cards. (AT&T will work, too, but only with 3G. You will only be able to use one AT&T card, not two. A 2G T-Mobile card next to a 3G AT&T card is possible. Verizon and Sprint are CDMA/CDMA2000 networks and do not work with the Fairphone 2 and most other European phones.)