Why a group is called allies in war?

This displays the phenomenon known as deixis. Using language to specify, by relating to a deictic centre (often myself, 'here' and 'now'): this car; my house; the Queen; there [pointing in some fashion]; today.

Men perhaps too often say 'the wife' / 'the missus' [in the UK] when meaning 'my dear wife'. 'The' is pressed into an unusual deictic sense (it's always deictic on some level, unlike the indefinite article). And here, 'the a/Allies' is pressed into the sense 'our allies', our group of allies.

This is covert propaganda (and inducing encouragement, esprit de corps); there are connotations involved. All comrades together, the alliance of all that is right. A holy alliance.

The name stuck, and has persisted:

The Allies of World War II, called the United Nations from the 1 January 1942 declaration, were the countries that together opposed the Axis powers during the Second World War (1939–1945). The Allies promoted the alliance as a means to control German, Japanese and Italian aggression. [Wikipedia; emphasis mine]

......

Remembering D-day and what the allies were fighting for [The Guardian, June 2019]

The word (in German) was even used by Germans:

THE first news of the Allied landing in Normandy came not from London or Washington but from Berlin.

"Die Alliierten sind erfolgreich in der Normandie gelandet" ("The Allies have landed successfully in Normandy") it read, and for 24 hours, while the Allies kept the lid on all news, it was repeated over and over around the world....

Wolfgang Straede, chief of the Berlin office of Europa-Press, an intra-European news agency based in Germany. As it happened, June 6, 1944, was to be his last day in his job: Denounced one time too many as a suspected anti-Nazi, he was fired.

But as one might imagine, the Allies' enemies did not usually use a translation of the English word to refer to the Western Alliance:

When it came to naming the Western allies, the Germans and Italians generally just referred to them by their nationality ... or sometimes 'Tommies' for the British.

[Chris Rhoden, Quora; re-ordered]

And note that 'the enemy' is similarly deictic. The Allies were refered to as 'der Feind' by the enemy. Australian War Memorial 'German First World War poster warning soldiers to be vigilant while on the telephone as the enemy could be listening:

• Der feind: hort mit! vorsicht am fernsprecher'

The British, being native speakers of English, got to determine what each side of the war was called in English, and in both cases, they chose the term “Allies” for their own side, presumably short for “Britain and its allies”. There was no reason for the Americans, also native speakers, to not go along with that. And the French weren’t in a position to argue with the powers coming to save them (again).

The Germans, Italians, Japanese, etc. presumably picked names in their languages that were presumably more flattering to their sides.

This is another manifestation of the same word functioning as a common noun when it is not capitalised, and as a proper name, standing for something more specific, in its capitalised form. Many religious institutions aspire to be catholic (with lowercase c), but Catholic (capitalised) stands for only one of them. In the United States, the supporters of one particular party are Democrats (capitalised), but many more people are democrats (lowercase), in that they generally support democracy. In the same way, any group of countries that are fighting together, on the same side, in a war (or, more broadly, people/organisations that are fighting on the same side of any conflict) can be called allies (lowercase), but only some particular such groupings named themselves Allies (capitalised). Of course, in a context in which the audience expects such a term to be used in its proper-name sense, it is wise to avoid using it in its common-noun sense.

In WWI I see that we have Central Powers and Allies. Both are groups of countries who fight together against the other group.

In WWII we again have Allies and Axis.

I can't understand why do we call a group allies, while both groups are composed of allies.

First, ally (n. and v.) comes from the Old French verb:

OED:

Etymology: < Anglo-Norman and Old French, Middle French alier, Anglo-Norman and Middle French allier, alyer, allyer (French allier ) (reflexive) to unite oneself or itself to (a person, group of people, or thing) (c1100), to unite, combine, or join in kinship, friendship, association, etc

Thus the people who were fighting on the English-speaking side were "allies" - they were united:

OED

**Ally (n.)

2. a. A person who helps or cooperates with another; a supporter, an associate; a friend.

and thus

2.b. A person, state, military force, etc., united or associated with another by league or formal treaty, esp. for political or military purposes.

In WW2, we had Britain and its allies, and Germany and its allies. For convenience, the English speaking British used "allies" to refer to their friends. But there was a need to distinguish Britain and its allies, from Germany and its allies - the Axis powers.

Q: So why "Axis"?

A: It comes from an obsolete meaning of the word, "axis" which is related to "axle."

OED: Axis

4.a. figurative. A central prop, which sustains any system (as [the god] Atlas was believed to support the revolving heavens). Obsolete.

1646 Sir T. Browne Pseudodoxia Epidemica 94 The Atlas or maine axis, which supported this opinion, was daily experience.

It then became a figurative use:

4.b. figurative. The relation between countries regarded as a common pivot on which they revolve; esp. the political association of 1936 (becoming in 1939 a military alliance) formed between Italy and Germany; later extended to that between Germany, Italy, and Japan; still later to that between other allied countries. Often used attributively, as Axis forces, Axis powers, and elliptically for such phrases, with consequent plural agreement. Also transferred, of any comparable association, or connecting common interest.

1936 Times 3 Nov. 15/1 The ‘Rome–Berlin axis’ is a conceit which has its momentary attractions.

1952 Economist 19 July 145/1 The Moscow–Peking axis.

1959 New Statesman 7 Feb. 177/1 The term ‘axis’ is looked on with disfavour here [i.e. in Bonn] as a reminder of the Berlin–Rome–Tokyo axis of the Nazis.