Would it make sense for a country to have an embassy for the whole of the EU instead of an embassy in each country? I was looking at this article and realized that the EU has its own diplomatic missions, although instead of having embassies they have delegations, not sure what the differences are exactly, but I am guessing they serve a similar purpose. Now, my question is whether it's even possible to have an embassy for the whole of the EU, and if it is whether it makes sense and what are the things you can't do with an embassy for the whole of the EU instead of an embassy for each country.
Would it make sense for a country to have an embassy for the whole of the EU instead of an embassy in each country?
While E.U. member nations have a border control union (for both people and goods entering the E..U.), they do not have a unified foreign policy on all matters.
Individual embassies for individual E.U. nations facilitates diplomatic communication and non-covert (and official cover) intelligence gathering by member nations, on matters outside of the scope of E.U. jurisdiction. Embassies of member nations likewise carry out many functions that are specific to that member nation such as applications for long-term visas, verification of citizenship for children of that nation's nationals who are born abroad, passport replacements, voting, referendums, declarations, apostilles (i.e. verifications that notarizations are authentic), and making extradition requests. Many treaties set up a system of country to country communication over treaty related matters that is designed to occur through embassies of the parties to those treaties.
Also, many countries have a mission to the EU in addition to their embassies in each country to carry out collective E.U. specific diplomatic agendas.
Furthermore, E.U. member nations are legally required (as noted in comments from o.m. and Matthew to this answer), under TFEU Art. 23 and the Consular Protection Directive, to cooperate in aiding each other's citizens diplomatically when they are in non-E.U. states. So, with respect to this consular assistance to E.U. citizens function, the E.U. country embassies and consular offices effectively do operate as a single, multi-location diplomatic mission.
Indeed, in some places (e.g. a group of diplomatic consular offices from several E.U. nations near Colorado Springs, Colorado), while E.U. countries maintain separate offices for themselves, the E.U. diplomatic offices in that place are in adjacent office spaces or buildings to each other. This is physically similar to having a single combined embassy with national wings within it at a combined embassy or consular office.
So, given that the E.U. nations coordinate as much as they do in their diplomatic missions why not consolidate them?
The ultimately most important point is that there are symbolic and quintessentially political factors at stake. It is a question of national identity, prestige, and perceived autonomy for the member nations and their citizens.
Most of the member countries in the E.U. don't want to be just sub-national states within a federal European nation. E.U. cooperation and coordination at its current high levels is already politically unpopular in many member nations. Since having a separate embassy is a signature symbol of being a top level sovereign nation and not just a subordinate bureaucratic unit within a federal country, the symbolic importance of each E.U. member nation have its own separate embassies should not be understated. This factor favors having separate embassies for each E.U. member nation even if utilitarian considerations alone would favor having a combined E.U. embassy system.
A single embassy can be responsible for relations with multiple countries, and ambassadors can be accredited to multiple countries at once (dual accreditation). This tool is commonly used in diplomacy with small countries.
Lots of countries can't afford to maintain separate embassies in many European countries. Those that only have a single embassy in the EU usually choose to locate it in Brussels. For instance, Tuvalu has a combined embassy to Belgium and delegation to the EU as its only diplomatic mission in Europe. Other popular locations for multiple-accredited diplomatic missions in Europe are London (especially for Commonwealth members) and Geneva (at the UN). You can find many examples of this if you browse the lists of diplomatic missions of various small countries (especially in Oceania and the Caribbean) on Wikipedia.
However, even when an ambassador is accredited to multiple countries, the relationships with those countries are still separate.
Embassies do serve the own citizens as well (passport replacements, voting, referendums, declarations, etc). Having an embassy further than the home country and behind multiple border crossings makes little sense for these functions. The idea is, the embassy should be much easier to reach.
For instance, if the passport is lost or just expired, the embassy could issue some temporary replacing document. But what if you need a passport to board the aircraft to reach them?