Can we own a bird pet which will in a cage? My cousin asked me to get one for her but I said : "Sorry, It's not allowed in Buddhism and we should care for them not in a cage even if cage were made of gold and must set them free and liberate them".
So may I know if it's allowed or not?
Instead of bad karma I felt myself pity and sorrowful for them. I felt so bad for animals if they're in trouble or something like this.
And isn't it same as slavery?
We making it as pet for our entertainment and not for caring for them.
So please guide me.
So may I know if it's allowed or not?

It's a Buddhist practice, in some countries, to buy caged birds (or sometimes other animals, like fish), in order to liberate them (i.e. to set them free, let them go).

I guess the purpose is to acquire merit, and that an idea behind it is that birds want their freedom.

An "unintended consequence" (of buying birds) is that people trap birds in order to sell them.

For this reason you might (or might not) want to refrain from buying birds, according to the same logic as being vegetarian (i.e. not buying meat in order to not encourage other people to kill animals): as you probably know, different schools of Buddhism hold different views on this subject.

IMO birds are a special category of the topic of, more generally, pet animals. Perhaps some people would keep fish (thinking that fish don't object to being kept); or "domesticated" animals like dogs, cats, cows, chickens, horses...

Buying dogs, incidentally, provides a financial incentive to breeders of dogs (i.e. it causes people to breed dogs). Some people suggest that you shouldn't buy a dog from a breeder, but instead "rescue" a dog from a shelter. When you rescue a dog, by the way, that's to keep the dog rather than to free it -- it appears that a pet dog would prefer to stay with its owner than to have to face the world alone.

Apparently there's an "unintended consequence" of getting dogs from a shelter: I read recently that some shelters are rich (from the donations they receive) and so buy puppies from breeders in order to have more to rescue! A way to avoid that, if it matters, might be to rescue an older dog (not a puppy).

It may be relevant to mention that animals in captivity sometimes live longer than in the wild -- they may be "safer", better fed, better cared for.

And isn't it same as slavery?

I don't know what goes on in the mind of a bird: I suppose they're not aware of slavery as a state-of-self, but are aware of a cage as a barrier-to-liberty.

"Slavery" (as a type of human relationship) would exist in the mind of the "owner".

From a human point of view, a good thing about not being a slave is that you can choose your master or boss (i.e. choose who you work for). If you don't like your current boss, you're at liberty to look for another.

Dogs seem to me more able and willing to communicate than birds. Pet dogs often have an opportunity to escape, and choose not to. The "leash" is often seen as being, and used, only for the dog's safety -- to protect the dog's safety rather than to protect the owner's ownership of the dog. I'm not sure that I see a pet dog as a slave: more of a friend, partner, or family member. OTOH some dogs are kept chained or caged: and it's easier to see them as "slaves".

I guess some people see a pet (dog or bird) as being like a friend or a kind of friend, or like a young child, rather than only like a slave.

Having (and looking after) a bird is a responsibility, which some people would prefer to avoid or to not seek -- if it's true that possessions are a burden, that (being burdensome) may be even more true of live animals -- but maybe that (whether it's seen as a burden or a blessing) depends on the attitude of the person.