The EU leadership is willing to grant some privilege concessions to the UK, trying to lure it into remaining within the Union. Why aren't the citizens of any of the other member states asked in referendum if they accept conceding these privileges to the UK, and, therefore, creating an unequal union?
Because granting these privileges is within the authority of the EU. Either the EU commission can do it themselves or they can ratify it through the democratically legitimized parliament. They do not need to ask the European public for every single change to the union treaty. Even forming the union in the first place wasn't ratified through a referendum in most EU states.
Doing an EU-wide referendum on such questions might be a theoretical possibility, but not required and most of all not practical. They need to make those offers now, because the UK referendum is imminent. When these concessions would only be offered under the additional condition that the other EU citizens agree, the influence of that offer would be greatly reduced.
Nationalist Movements in other EU Countries aren't 'stoking the fires'
There has been some discussion of EU exit from other member countries but at the end of the day the Nationalist movements that are spreading and winning elections frequently understand that EU membership ultimately has more benefits to them and economic stability than not.
The whole idea of the EU started in the 50's to create closer economic dependencies between various countries in order to prevent war and disharmony between neighbors. This is still true today within the EU as well as globally. Most moderately educated politicians understand this paradigm and therefore resist the urge to stoke the flames of Nationalism during their elections and campaigning.
Outside of politicians stoking exit desires the citizens of the EU seem to have a mixed reaction to the idea of the EU as well as closer ties to the EU and the transfer of national power to the EU governing bodies.
Typically countries that are suffering financially, from high unemployment that is still hounding some countries from the financial collapse, have citizens that are squarely not in favor of EU membership. Spain and Greece come to mind. Other countries, even with strong Nationalist movements (other than France), seem to have a slim or high majority in favor of continued EU membership. Former Warsaw pact countries are definitely more inclined to have a populace that is firmly in favor of the long game of EU membership coupled with the economic development and open markets that will make their economies better, Poland is a great example of this.
It's a mixed bag of demographics. The bottom line is that there aren't many credible politicians that are trying to rally their party and the associated electorate for a local referendum. The EU has been a good deal for many countries. Other wealthier members have had to carry a lot of economic weight regarding bailouts, reforms and dwindling employment numbers. Great Britain being one of them.
Here's a great opinion poll reflecting various demographics that support/oppose EU membership.
Here's another study regarding public opinion in the EU.
EDIT: Post Question Edit Response
The UK is not asking for anything special. The concessions they are asking for and have received would apply to ALL EU member states. This, of course, would not weight or provide favoritism to the UK.
Here are the concessions.
Four year delay for EU migrants wishing to claim in-work benefits, such as tax credits, or seeking access to social housing
Stopping migrants claiming child benefit for dependents living outside the UK
Removing migrants from the UK after six months if they have not found work
Restricting the right of migrants to bring non-EU family members into the UK
Stopping EU jobseekers claiming Universal Credit
Speeding up deportation of convicted criminals
Longer re-entry bans for beggars and fraudsters removed from the UK
Stopping citizens from new EU entrants working in the UK until their economies have "converged more closely".
Extra money for communities with high levels of migrants (source)