Is the Age of Empires Series a 4X Game?

See Wikipedia's article on the subject, specifically the Difficulties in Definition section:

While many computer strategy games arguably contain a similar "explore, expand, exploit, exterminate" cycle, game journalists, developers and enthusiasts generally apply "4X" to a more specific class of games, and contrast 4X games with other strategy games such as Command & Conquer. Hence, writers have tried to show how 4X games are defined by more than just having each of the four Xs. Computer gaming sites have stated that 4X games are distinguished by their greater complexity and scale, and their intricate use of diplomacy beyond the standard "friend or foe" seen in other strategy games. Reviewers have also stated that 4X games feature a range of diplomatic options, and that they are well known for their large detailed empires and complex gameplay. In particular, 4X games offer detailed control over an empire's economy, while other computer strategy games simplify this in favor of combat-focused gameplay.

The difference is somewhat subtle and hard to articulate cleanly. It's often easier to point to games 'in' and 'out' of the cluster and learn the boundary from there than it is to explain why any particular game is in or out of the cluster; for AoE specifically, the argument is "it plays more like StarCraft than like Civilization."

There was an old game called Empire.

From it descended the games Civilization and Dune 2.

Dune 2 is the grand-daddy of the RTS genre, and Civilization is the grand-daddy of the 4X genre.

Games that are "like" (the original) Civilization, and "like" the more famous games that where categorized as 4X, are categorized as 4X games.

Games that are "like" Dune 2, and "like" the more famous games that where in turn categorized as RTS, are RTS games.

This holds true even before the labels 4X and RTS where defined.

They are similar games in some respects; but their definition is determined by their difference. Their labels describe the genre, they do not define the genre. The genre exists independently of the label; it is influenced by the label, but it is not "anything that matches the definition is the genre".

For other examples of where this happens, examine the genre "urban fantasy". It does not apply to any book containing fantasy elements in an urban environment; rather, it is defined by the archetype of the books usually with a female main character, with tattoos, set in a pseudo-contemporary setting, with a romantic triangle subplot (usually involving vampires or werewolves), etc.

A popular exemplar appears. Copycats develop, with variation. The clump of copycats and the exemplar examine a space of ideas; the clump becomes the genre. Latter copycats expand, migrate or refine the genre clump. Sometimes a sub-genre develops, or splits off, with its own pattern of ideas.

What does Age of Empires distinguish from e.g. Civilization V regarding those points (explore, expand, exploit and exterminate)? Civilization V appears to be a very famous X4-Game and I am pretty familiar with it.

Age of Empires, Starcraft etc are very much "all-action" style real-time strategy games. The phases for exploring, expand, exploiting and exterminating do exist in such games but are very simple, and often quick to perform. The main aim of the game is to wipe out the enemy. The complexity will be in your constant handling of units real time over making thoughtful, planned decisions. Sure, you make decisions, and there is a strategy, but you need to make the decisions fast and you see the outcome very quickly.

In games like Civilisation the phases are far slower, and far more complex. There are often multiple paths in each phase. Diplomacy, as you may know, has a greater bearing in games like Civilisation. 4X games will often last a lot longer than something like Age of Empires. Another difference is the scale of a battle. In Age of Empires you have a small regional map whilst Civilisation will be an entire world or continent.

If you read the wiki article in full you will probably find that complexity and time are key points raised regarding what defines a 4X game.

I read a bit, mainly on here:

The TL;DR version of what I found is that its about complex ways to win and diplomacy.

AoE is one army on each side fighting and the win condition is destroying the enemy. Diplomacy is mainly peace or war.

Civ is way more complex such as having unit upgrades and more ways to win. Diplomacy goes way over just peace or war.

Similar to what others said, but I think a more clear reason is that AoE is pretty heavy into the RTS camp. The only thing that could separate it from other RTS games is that it has diplomacy, but this diplomacy is severely limited and IMO does not set it apart from other RTS games.