Council of Elrond

Playable on a site. A company moving to this site this turn must do nothing during its site phase. A ranger in such a company may tap to cancel this effect, even at the start of his company's site phase.

Company A moves from Himring to Bree, then with help of Mater of Esgaroth, it moves to Mount Gram.
Then company B moves from Old Forest to Bree.
During M/H phase of company B, River is played on Bree.

Is the company A affected by the River played on Bree during the M/H phase of company B?

I would say it is played on the site. So it works cross companies. The site needs to be removed to get rid of the River. And since another company is moving to a site on the board, it does not get discarded. River stays and affects the 2nd company.

just like an on-guard played on 1st company would stay and affect the 2nd company

CRF, Card Errata and Rulings, River wrote:Card Erratum: Should read "Playable on a site. If a company that has moved to this
site this turn does not tap a ranger, it must do nothing during its site phase." You have
until the beginning of the site phase to tap a ranger, and you may tap the ranger at the
beginning of the site phase without entering the site.

Text of River at https://meccg.com/cards/
does not reflect the change precisely.

River wrote:Playable on a site. A company moving to this site this turn must do nothing during its site phase. A ranger in such a company may tap to cancel this effect, even at the start of his company's site phase.

The original text of River uses the phrase "If a company without ranger that has moved to this site this turn [...]" (Lidless Eye), or "A company without ranger that has moved to this site this turn [...]" (The Wizards).