Well if we leave strict definitions aside for a moment and go on the general qualities and sentiment of a “nation” then I would be inclined to believe that yes the EU is in some form a country.
Let’s take a look at what goes to make a country. In a loose sense it is a geographical region that is unanimous in its intent to have cooperation between its peoples both economically and socially.
It abides by a common set of laws and rights/liberties or freedoms that it offers all of its people equally.
When we think of the EU we think; freedom of movement, freedom of labour, a common tax/customs area for goods and services, a more or less common currency and language of commerce, A standardised qualifications comparison system, and a governing body that offers provisions based on need so that the whole can be “brought up to speed” - that is to say established on a level playing field with one another in a developmental sense - the weakest countries often considered second world or even third world have in a matter of decades been accelerated into first world conditions/ quality of life.
People in the EU generally consider themselves as a “European citizen.” The EU is a place of common interests and community that has sought to expand its territory and “unite” people of different areas under one geographical/ social/ legal/ financial and political entity.
It has many parallels with a country. Allow me to make note that the term “country” is relatively arbitrary in definition; by regard of religion (most countries are multi-denominational already), Socioeconomic (many countries have distinct socioeconomic regions (think north Italy vs the Mezzogiorno), linguistically (many countries have more than one officially recognised language (Ireland has English and Irish, Spain - Galician, Catalan, basque and castellano), India has 20+, as well as dialects within a language - Italy for example. Currency - several nations have multiple currencies: Cyprus, jersey islands, Bhutan and Cuba are typical examples. And politics - the USA is highly partisan as well as Britain and many others with distinct political sub populations within one nation.
So it’s clear what goes to defining a country is, at best, difficult and vague. I would look at it from a historical progression. Tribes were the first communities. Every country has had a diverse array of tribes, clans or families which led their territory, then came kingdoms of which Italy had dozens upon dozens - each very unique, these Types of regions evolved into unified countries which then unified into larger entities such as the EU. The term tribe, kingdom, nation or union are somewhat synonymous in that they are symbolic of human cooperation and willingness to identify as a group with a mutually agreed border. An ever expanding “us” and ever diminishing “them”.
If we were to encounter life on other planets the scale would once again inflate and we would likely identify as a global nation/country in an intergalactic set of “countries” or territories.