At least in a Western context people widely agree upon the idea that you have to do something voluntarily to be responsible for it. If you sign a contract voluntarily you are bound by it but if someone forces you, at gunpoint, to sign it you are not bound by it. The law and moral generally agree on situations like that.
And it goes even further, most of the time you must have intention to do something to be morally responsible for the consequences of it. If you stumble on a train because it makes a sudden stop and falls on someone who is injured, you are, generally, excused and not convicted for abuse, assault or similar - and most people think this is reasonable.
With all this in mind, how do people motivate that children when they turn 18 or something like that no longer can expect their parents to support them?
The parents voluntarily and intentionally (they intentionally had sex, if the child was unplanned it is still a well known fact that having sex can lead to "accidents" and they obviously didn't terminate the pregnancy so in the "normal case" the child "happened" intentionally and voluntarily) had a child. The consequences of that decision is a human being that will need food, clothing, shelter and so on for the rest of her life.
However, by the logic from the first two paragraphs, the child is not responsible for her situation and the consequences of it because she didn't do anything to end up needing all this stuff for the rest of her life.
In fact, we are more generous towards our pets than our children in this respect. We never expect dogs and cats to provide for themselves when they reach a certain age. Instead we realise that getting a cat or dog means that we have to support it with food and shelter for its whole life (yes you can put down an animal but generally doing that to a healthy animal is frowned upon and most people would feel immoral putting down an animal because we don't want it anymore).
Two final remarks:
- You can't own a human, at least not unless it voluntarily sold itself.
- Suicide is, IMO, not a reasonable answer. We are biologically programmed to survive (Eat poison and you will throw up. If you are so determined that you can hold your breath until you pass out your autonomic nervous system will save you as soon as you no longer can control your breathing with will power.). It would have been different if our biology were indifferent between life and death.
So how do you morally and ethically motivate that a child must take responsibility for her parents' decision just because she turns 18 (or something like that)? Why aren't parents obliged to support and provide for their child's basic needs for the child's whole life?