Yes, this is called a "many to one" matching. Many to many matchings also exist.
Note that not all many to one matchings are a two-sided matching market. E.g., if I buy several oranges, that is not a two-sided matching, I am the only one with any input w.r.t. the matches.
Similarly, entries in different tables of a database can be matched one-to-one, one-to-many or many-to-many. However, there is no market here at all, just a mapping.
Some US municipalities use a many to one matching algorithm to match students to high schools. Since the preferences (rankings) submitted by students as well as schools are considered, this is indeed a two-sided market.
The US National Medical Resident Program also does this.
The New York City High School Match
Initial discussions focused on whether the
medical match was a good model for New York
City schools, or whether another kind of clearinghouse might be more appropriate. The medical match applied to schools would be a twosided model in which both schools and students
have preferences, with the object of implementing a stable assignment, that is, an efficient
assignment such that no school and student not
matched to one another would both prefer to be.
Thus, the question was, are the students the only
real players in the system, with choices by
schools merely a device for allocating scarce
spaces? If this were the case, there might be
appropriate one-sided clearinghouse models in
which only student preferences determine efficient allocations (cf. Boston Public Schools;
Abdulkadiroglu et al., 2005).
Two things convinced us that New York City
schools are a two-sided market. The first was
that schools withheld capacity to match with
students they preferred. Stable assignments
would eliminate the main incentives for this.
Second, discussions indicated that principals of
different EdOpt schools had different preferences even for students with reading scores in
the lowest category, with some schools preferring higher scores and others preferring students
who had good attendance. If schools have different comparative advantages, allowing scope
for their preferences seemed sensible. Also, the
fact that school administrators gamed the system indicated they were strategic players.