Conservative pastor and OT scholar Gordon Hugenberger of Park Street Church in Boston argues that the biblical view of marriage is less formal and ceremonial than we view it today. It had more economic overtones, less romantic ones (at least at the outset), and generally speaking, engaging in sex automatically created the marital covenant. (See his dissertation published as Marriage as a Covenant and his sermon series on marriage -- MP3s here -- and subsequent Q&A, which calls out the titles of the sermons in the MP3 link.)
In short, I think he would say a male-female sexual relationship is viewed as itself creating a marital bond, whether formalized publicly or not, and dissolution of that relationship would amount to a divorce or breaking of the marital covenant. (A non-sexual partnership or an otherwise prohibited sexual relationship would not create such a bond.)
As a qualification on this, he notes that the issue of "premarital" sex was far less common than it is today in our wealthier, leisurely society with its delayed marriage age:
Throughout the entire ancient Near East, men and women normally got
married in their mid-to-late teens. For this reason, there were very
few individuals around who were old enough to be sexually mature, but
who were not yet married or at least "pledged to be married."
(Engagement or "betrothal" gave couples the legal status of marriage,
even though their marriage was not yet consummated.) This is why far
more attention is given in the Bible to the problem of adultery than
to the temptation of premarital sexual promiscuity. There was very
little in the ancient world comparable to our practice of dating.
As for history, conservative Catholic writer Ross Douthat argues that the Western, Judeo-Christian-derived ideal for marriage
holds up the commitment to lifelong fidelity and support by two
sexually different human beings — a commitment that involves the
mutual surrender, arguably, of their reproductive self-interest — as a
uniquely admirable kind of relationship. It holds up the domestic life
that can be created only by such unions, in which children grow up in
intimate contact with both of their biological parents, as a uniquely
admirable approach to child-rearing. And recognizing the difficulty of
achieving these goals, it surrounds wedlock with a distinctive set of
rituals, sanctions and taboos.
Marriage in this understanding is "not some universal, biologically inevitable institution" but rather is
a particular vision of marriage, rooted in a particular tradition,
that establishes a particular sexual ideal [and] it was the Western
understanding. Lately, it has come to co-exist with a less idealistic,
more accommodating approach, defined by no-fault divorce, frequent
out-of-wedlock births, and serial monogamy. [emphasis added]
On the whole Christian tradition has supported the idealistic vision thoroughly, until the mid to late 1900s when a "less idealistic" vision found some supporters among the more liberal members of various branches of the church and the culture at large. It has consequently seeped into more conservative churches in various ways (e.g., in how the church treats divorce), though most still hold out against some trends, such as premarital sex and cohabitation.
As for your updated question about mainline denominations' views: the mainline churches are hardly a monolithic whole, so generalization usually leads to some inaccuracies in comparing between groups (say, the UCC and the PCUSA). The mainlines tend to be relatively "big tent" operations with a wider variety of views than a more theologically conservative denomination, though there are often still a conservative subset within their tents. As a rule of thumb, the more conservative the group as a whole, the less variance there will be in their views.
Views on marriage can and do range widely even within on particular mainline group. Some progressives have abandoned any semblance of a traditional or biblical ethic in setting the parameters for marital relationships, preferring some less stringent "ethic of love" (or whatever). Votes on such matters at the denominational level are often very close, indicating that there is not a strong consensus within the group. Even when a view is adopted, there can be a large minority group within the denomination that does not agree with it.
All that being said, to my knowledge no mainline group officially recommends cohabitation, etc. They still prefer to see a public commitment in the form of marriage, even if their definition of marriage includes differences from the Western ideal described above, such as easy divorce or same-sex couplings.