Tendency of using pronouns 'she/her' when talking about a random person

Reading different specifications and manuals I've noticed that more often and often pronouns she or her are being used when some unknown person's behavior is described.

For example: "when user opens the first screen, let she be notified about something" or "let user input her address". Actually, I've seen more versatile examples but cannot remember anything nice now.

Is it a last-years tendency? Or is it something only UK-specific? Are there any rules about it? Or, maybe, is it applicable only for written document/spoken language?

I'm not an informant (as you've already guessed) and it's very important to me how should I write in official correspondence with British/US customers and not to hurt their feelings?

The pattern of using she for an unknown referent in formal writing is a contemporary attempt to balance out the perceived sexism of generic he. However, most English speakers find generic she to be at least somewhat marked or surprising, so many of the works that I've seen that use generic she specifically mention that they will be doing so in their introduction. Generic she is also sometimes used to disambiguate between two participants in an action, such as in a game manual where one player is consistently called she and one he.

However, there is a third alternative which may be preferable for many occasions: singular they.

When the user opens the first screen, they should be notified of their options.

This is the form that most English speakers will use in speech, and it should be acceptable in all but the most formal writing. There is some prescriptivist carping against this form, despite the fact that it has centuries of precedent and divine sanction. These objections can be safely ignored.

This seems to be a growing trend in written texts of late. I believe that the rationale must be a desire to use gender neutral pronouns and balance out the decades (centuries?) of male only pronouns being used throughout literature and written texts.

As long as you are fairly consistent and balanced using gender specific pronouns, there should be no issues. This is assuming that you are not specifically aware of the gender of the recipient.