If I understand it right, correct term for such situation is bilateral oligopoly.
Hendricks and McAfee (2007) offer a theory of bilateral oligopoly. They consider the example of the wholesale gasoline market on the west coast of the United States, which is composed of a small number of large sellers and large buyers who compete against each other in the downstream retail market. They are specifically interested in the effect of a merger of vertically integrated firms on the whole sale and retail markets. They discuss price and cost formation. They also offer a review of the literature.
The extreme case is bilateral monopoly with one buyer and one seller. Tirole's Industrial Organization textbook (pp. 21-25) offers a nice discussion of how this market works and in particular price behavior under bargaining and contracting.
Yes, that is correct. In an oligopoly there are many buyers and few sellers, so buyers are more passive when it comes to prices. The few suppliers can also engage in price discrimination because there are many buyers with a higher demand level for a given good. However, you are correct in that what you described is a bilateral oligopoly. You can relate this to the fact that the few buyers have NEGOTIATING POWER. For example, Boeing is a large purchaser of jet engines, so they do have negotiating power over the three major jet engine suppliers because they are one of the few consumers in the market place. That’s how I personally relate the idea and commit it to memory.