I am not defining friendship. It's easier to say what is not friendship. In most aspects there are some signifiers of friendship. What language do you use, how do you treat people, what do you feel. People are not going to use formal language in a friendly conversation. People experience compassion towards their friends. People do not treat each other as property if they are friends.

There are several other things friends do not do to each other.

But an important one is that friends do not rule each other. They don't think of each other as of subordinates. Doesn't it make my idea a type of anarchy?

In friendship there is a mutual understanding.

These are the definitions of 'anarchy' given in some dictionaries:

A lack of organization and control in a society or group, esp.
because either there is no government or it has no power:

A state of disorder due to absence or non-recognition of authority
or other controlling systems.

Absence of government and absolute freedom of the individual,
regarded as a political ideal.

If you describe a situation as anarchy, you mean that nobody seems
to be paying any attention to rules or laws.

A state of lawlessness or political disorder due to the absence of
governmental authority.

Now you may check these definitions giving importance to the bold words without forgetting the term, 'mutual understanding'. Can you still say this is a type of anarchy? I believe you can't. If so, what you doubted is wrong.

This doubt is because we normally don't need to discuss this type of a weird state and so we don't need to name it. Actually this is a new state. Though there is no physical presence, an invisible governance is happening and it is by 'mutual understanding' among people. So we cannot categorically use the term anarchy.

Anarchism and politics

'Anarchism' as a matter of ordinary discourse is mainly tied to politics. An anarchist society is a voluntary, non-coercive social aggregate. There can be no legitimate government that exerts or threatens to use coercive power against its citizens. Political authority - the entitlement of a government to obedience - is inherently conflictive with individual autonomy, which is seen as an indefeasible value. Autonomy is exercised in precisely the voluntary social aggregates mentioned just above.

This concept of anarchism has to be outlined because it is pretty much, or so I believe, the main sense of the term in ordinary discourse.

Islands of anarchism

This doesn't mean, though, that even in a society where a government - the state - exerts or threatens to use coercive power against its citizens there cannot be social interactions from which coercion is absent. In a commune or a friendship, interactions can be marked by voluntary, non-coercive relationships. In this sense they are islands, small units, of anarchism within an overall coercive society.

**To the extent that your friendships are voluntary and non-coercive, they embody an element of anarchism. I put it this way because I don't think that wanting everyone to be a friend with everyone is necessary for anarchism. In an anarchist society, voluntary, non-coercive relations would hold even between people who were indifferent or antipathetic to each other. Nor, more to your point, is it sufficient. A and B might be friends yet A reject and B accept a supreme coercive power. Friendship dictates politics only up to a point because friends can have different views about the means - the instrumentalities - required or most likely to maintain civil peace and social justice. Whatever our friendship, you may think the state is an abomination while I see it as a means, even if a necessary evil, to promote my social ideals.

Foucault

We have learnt from Foucault that power is present in all social interactions. Friends may exercise power over each other even in ways neither or none recognises. But I don't think this nullifies my point. Anarchism focuses essentially and antagonistically on the supreme coercive power of the state; and a friendship typically avoids any counterpart to this.

Broadly speaking, Foucault was not interested in the kind of power that anarchists traditionally oppose :

a group of institutions and mechanisms which ensure the subservience of the
citizens of a given state . . . [nor] a mode of subjugation, which in contrast to
violence has the form of the rule ... [nor] a general system of domination exerted by
one group over another. (Michel Foucault, t, The History of Sexuality: Vol. 1 An Introduction, London: Allen Lane, 1979 : 92.)

Rather, Foucault holds a 'capillary' notion of power:

Most scholars emphasize the contrast between Foucault's relational conception of power and the more prevalent understanding of power as a property which can be possessed. They stress his sensitivity to the fluctuating network of power relations, his development of the "capillary" conception of
power - a micro-power which permeates all social strata producing and thus
constraining subjectivity - and his notion of bio-power, which he considered
to be an indispensable element in the development of capitalism. Some under-
score his constant endeavor to problematize the "normal," praising Foucault's
success in showing that phenomena which society deems permanent,
inevitable, and universal, are but a specific period's fabrication. Moreover,
commentators tend to agree that Foucault has opened a new path of critical
inquiry. (Neve Gordon, 'Foucault's Subject: An Ontological Reading', Polity, Vol. 31, No. 3 (Spring, 1999), pp. 395-414: 397.)

Dense as this quote is, it sends us in the right direction. Foucault does not see power as a property possessed by individuals or social units such as politicians, bureaucracy and the security forces or Hobbes' sovereign - at least not the power he's interested in - but as relational. A network of interpersonal relations, over which no-one has sole control and of which nobody has complete knowledge - creates us as 'subjects' and provides the language and concepts by which we 'construct' ourselves as persons or agents. It creates our self-understanding.

I am not entirely happy with this language and do not offer to appraise Foucault's idea of the subject - a shifting idea, I should add. My point is only that the anarchist is not as such concerned with Foucault's idea of how 'subjects' are created. The anarchist is concerned with and opposed to supreme coercive power, the entitlement of governments to obedience, and with the right environment in which for individuals to exercise their autonomy. Foucault takes a different (perhaps subtler) view of things, especially about the formation of the individual. But an anarchist need not agree with him; and this answer proceeds, respecting as it should the OP's question, from the viewpoint of the anarchist.