As there are nonsexual marriages/deep relationships among mono folk, there can certainly be such among poly folk.
And doing poly doesn't require a full dance card of partners. Many folks are quite happy having only two romantic ties and have no interest in any more. There are stable vees where one person has two partners, the other two only have the hinge partner, and everybody's happy and fulfilled. There are stable triads where the three are involved with each other and everybody's happy and fulfilled. And quads and so on and so on and so on.
You only have to have as many relationships as you desire and can handle to do poly. If one other is all you want, that's all you have to have.
When speaking of various forms of non-monogamy...it ain't poly if you're just fucking around.
While polyamory, open relationships, and swinging are all distinctly different approaches to non-monogamy, they are not mutually exlusive. Folks can, and some do, engage in more than one of them at a time--and it's all good.