Actually you are wrong. Zika virus was isolated from culex and anopheles. The answer why aedes is primary vector may be the fact that it is very aggressive.
There are several biological factors of vector which suits to the pathogen like dengue virus, They have adapted in Aedes by continuous process of picked by mosquitoes.
There is a mid gut barrier for most of the pathogen coming in mosquitoes through human blood those pathogen who are able to cross barrier they try to reproduce in the cells of mosquito and then when there incubation period is over they migrate to salivary gland for infectivity. dengue virus has adapted over Anopheles sets of proteins which may be required for its growth.
Factors :
Longevity of mosquitoes , Incubation period of virus
Mid gut barrier proteins (same genus of Aedes may not transmit due to presence of absence of particular protein sets midgut )
Required proteins for development in species of mosquitoes
virus adaptation in particular species
vector biology etc and many other factors
its just fine tuning of virus with a specific host
To comment on Lucasz's answer, if you read that article carefully you will see the authors caution against interpreting Zika virus-infected with Zika virus-transmitted. The known vectors of ZIKV are predominantly Aedes africanus and Aedes aegypti, amongst several other aedines. For this reason, it was wrong to say that Naresh was wrong.
However, it might also be wrong to say that Anopheles mosquitoes cannot transmit viruses. There are reports of experimental transmissions (eg An. plumbeus and West Nile virus), and also some field evidences that different Anopeles species may transmit Rift Valley fever virus (eg in Madagascar).
I think it needs to be noted that o'nyong-nyong virus is the one medically important arbovirus that can be transmitted by anophelines and culicines. In practically all other instances, culicines are the exclusive or nearly exclusive vectors. In general, the main genera involved are Aedes and Culex. An arbovirus may have been isolated from numerous species but may reflect the presence of an undigested infected blood in the midgut or a salivary gland infection barrier, for example. Susceptibility must not be confused with the ability to transmit.
The musquito of aedes has some 34 kDa protein in saliva which has affinity to bind to viral antigens . probably this might be reason . but on literature search other species of musquito may also carry such receptors or proteins.
Vector-parasite interaction is a result of the long journey of coevolution of both vector and parasite along with the host that took millions of years... During coevolution host (primary or secondary) have developed means to exclude the parasite, whereas parasites developed means to use the host proteins for its own survival...
Regarding specific vector would transmit specific parasite: I take refuge to Gaussian exclusion principle that states, when habitats of two species overlap both the species compete with each other and one species excludes the other.
In nutshell what we are observing is a result of the complex evolutionary/ecological processes...