One common romantic desire is the wish to be owned by one's lover, to be his or her property. Mutual ownership only goes so far, because at the core it is a kind of power dynamic where the "owner" stands for safety and responsibility, while the owned can relax and feel small in a good way. Often includes Freedom from Choice.
To avoid Unfortunate Implications, a Property Of Love dynamic is often justified in one of several ways:
A. Simply Romantic: Hoping that its cuteness will avert any hatedom.
D. Cultural Relativism: The dominance is based on gender or race or similar. However, the story takes place in another time or on another world, and this is empathized in a way that hints that the Values Dissonance is intended — the reader may be troubled, but the characters themselves are fine with it.
Regardless of justification or lack thereof: When this trope is played straight, it is about a heart freely given, without any coercion or big deceit/manipulation. However, in some stories a kidnapping can go from Stockholm Syndrome to genuine mutual love.
Being a Love Trope, the Property of Love trope in itself has no inherent connection at all to tropes such as Sex Slave or Happiness in Slavery. However, these three tropes can overlap in at least two ways. First, as a benevolent consensual BDSM game, playing with the concept of slavery. Second, as the previously mentioned Stockholm Syndrome version growing into the real thing.
The Inverted Trope, avoiding relationships out of fear of becoming "owned", is largely a Gendered Trope. In old works, it's mostly about men trying to avoid being captured by a woman. In contemporary works the situation is reversed, centering on women striving to be independent from men. Both versions are largely Truth in Television, due to changing social structures: A few decades ago, men typically had most to gain and least to lose from being single, and it remains true to this day.
Note that Bob claiming romantic ownership over Alice (or vice versa) doesn't make her his property in any way. It can be an expression, or it can indicate that he's a Stalker with a Crush or a Bastard Boyfriend.
In Guilty Crown, Shu and Inori have a fair bit of this dynamic going on in their romance arc, but notably, it doesn't go as far as Inori being willing to leave her Organization and its leader (who liberated her from the lab she was created in) when he asked her to. At first, it isn't clear how much of this was an act she was instructed to put up to keep Shu and his powers available to her boss, but it becomes a genuine sentiment as the show progresses. Note that for Inori, being "his" also means always staying on his side and continuing to support him - which he needs quite badly.
Rosario + Vampire: An anime-exclusive character called IjuuinKotaro has the ability to magically charm any female. He successfully charms Moka, Kurumu, Yukari and Mizore, and for a whole episode they become his obedient slaves, completely in love with him. Moka and Mizore even affectionately call him 'Master' while under the spell.
Beautifully parodied in Wonder Woman #600, where Diana and Power Girl have a seemingly very uncharacteristic conversation about how Karen needs to understand and accept that "you belong to him", before The Reveal that they're talking about her cat.
Beautifully averted in Hobbit-Fanfic The Toymaker and the Widow, as Bofur makes it perfectly clear that his love interest is nobody's property. Also played straight with Dwarven marriage custom, according to which a widow is still considered the property of her deceased husband, and, thus, cannot remarry.
Inverted in the film version of Breakfast at Tiffany's: When Paul tries to talk Holly Golightly out of going to South America because she belongs to him and he to her because they love each other, and she insists that she doesn't, and won't, belong to anyone, and that's part of the reason why she has to leave.
Holly: People don't belong to people.
In Captain Blood, the trope gets inverted for both genders. Blood gets sold as a slave and bought by his eventual love interest. Later she gets taken prisoner and is sold to him in a similar fashion. They both find the experience to be humiliating in a bad way, and a total turnoff to the point of stopping them from admitting their feelings for each other. It is only later, when they are both free, that their love for each other can bloom.
Secretary is a mixture between drama and Romantic Comedy, revolving around the BDSM version of this trope. The main storyline is about the main character wanting to be truly owned by her boss.
In The Sound of Music film version, the recap of the song "Seventeen Going on Eighteen" has the clause, "and you belong to him." At least one stage version performed during The Nineties changes the clause to "and you belong with him."
True Blood has this as one of its basic premises. Sookie have given herself to Bill out of love, without being hypnotized like so many other fangbangers. Also, being his makes her off-limits to other vampires.
The same pattern of vampire owning human is consistent in the series, and it's often a female vampire owning a male or female human. However, these relationships are rarely as happy as the one between Sookie and Bill. In the second season, one guy belonging to a female vampire is so desperate for her love and blood that he'd do anything for it... including betray her.
In Doctor Who, it's revealed that the TARDIS views the Doctor as this. She points out it's mutual, as she's his sentient starship/time machine (which he boosted from a Gallifreyan museum), but he hadn't thought about it in quite that way before:
The TARDIS: Then you stole me. And I stole you.
The Doctor: I borrowed you.
The TARDIS: 'Borrowing' implies the eventual intention to return the thing that was taken. What makes you think I would ever give you back?
In Gor, this kind of love is based on a very violent and misogynistic culture where males are supposed to have power over females. The author relies heavily on justification D, constantly pointing out that the stories are taking place on the planet Gor and not the planet Earth. After a while, the descriptions of how the sheep of Gor grazes on the fields of Gor to create the wool of Gor gets more annoying than the actual storyline.
In further evidence that not only are tropes not bad, but modern Western assumptions about gender may not even be that accepted in the West: there are literally thousands of separate storylines invoking justification D, spread throughout shelf after shelf of the romance/erotic-romance sections of bookstores...and written mostly by women. These stories largely fall under another purview of similar murky morality in fiction: They are fantasy, and incidentally romantic interpretations of the trope, and do not necessarily reflect the reality, in all its non-romance, of the situation.
Jean-Claude from the Anita Blake tends to form these sorts of relationships with his humans and were-creatures, including Anita, later in the series. Even when the master/servant relationship isn't sexualised (for example, between Jean-Claude and Jason), it tends to involve high levels of devotion ( and possibly also Stockholm Syndrome).
Played with a lot in The Red Vixen Adventures, Alinadar wants this kind of relationship in large part because she spent most of her life as a slave, and she developed a serious crush on her second mistress The Red Vixen, who unfortunately was straight, but who thought she could set her up with Sallivera. However Salli didn't like the idea and avoided getting serious with Ali until she got her head together and could handle a slightly more equal relationship.
For an example from the days when it was considered simply romantic, there's the 1930s jazz standard "Body and Soul": "I'd gladly surrender myself to you, body and soul". While the "owning" thing was unproblematic in those days, there still was a bit of trouble about the song when it first came out. Not because of the "owning" thing but because some people thought it was a bit risqué for a woman to sing about surrendering her body to a man.
Romantic ownership is pervasive in pop music. The phrase "I Belong to You" is the exact title of many different songs by popular artists like Lenny Kravitz, Anastacia, Brian McKnight, Toni Braxton, and Whitney Houston, most of which also mention "you belong to me."
In Romeo and Juliet, Paris, Juliet's fiancé, regards Juliet this way; when she cries, he says, "Thy face is mine, and thou hast slandered it." Note, however, that this kind of language was commonly considered romantic and sweet at the time, a reference to the unity of husband and wife, and doesn't necessarily justify playing Paris as a Jerkass.
If you take The Taming of the Shrew at face value rather than trying to read subtext into it, this is where Kate ends up.
This trope, played for romance, is basically the point behind Kissed By The Baddest Bidder, which begins with the protagonist finding herself auctioned off to the highest bidder at a secret Black Market auction. Exactly how the result plays out depends on which of the guys ends up taking charge of her, but it's always ultimately depicted as romantic when she decides that she's okay with the man she's in love with calling her his property.
In the BDSM themed webcomic Sunstone there are aspects of this to Lisa and Ally's relationship. Lisa gets a tattoo reading "Property of Allison" and her wedding ring has the inscription "Forever Mine" on it's interior. Lisa also likes to wear her collar in public under a scarf to remind her that the previous night she "was hers". Marion on the other hand was keen to enter a situation where she was submissive to Alan 24/7, wore her collar and referred to Alan as "Master" at all times.
Quite literal in Fairy Dust, where orc patriarchs buy their wives from allied clans. While they are considered his property, female orcs are just as combattive as the males and they outnumber him twenty to one: earning their love is the wisest option.
Mackenzie, the protagonist of Tales Of MU, is DEEPLY submissive, and gives herself to Amaranth (a nymph) in this way as a Type A. The ogres in said work also have a strict chain of people-ownership, romantic and otherwise, which leads to Mackenzie being treated like a simple object because she's too far down the chain of sexual dominance.
This short manhwa plays around with this trope. A presumably goblin-like mafia boss buys an elf slave girl and treats her with kindness. She eventually feels very comfortable living with him.