There's no work done for a person climbing upstairs because the energy is converted to PE within system only. The person is the system.
How true is the above statement?
The above statement is not correct.
First of all, you need to work against the force of friction while climbing stairs.So the energy is not entirely converted to PE.Rather a portion of it is dissipated.
Secondly, even if we leave out friction, the basic flaw of the statement lies in the part: " the energy is converted to PE within system only. The person is the system. ".
The concept of potential comes from the fact that work is done to bring a point mass from infinity to another point in presence of some field.So potential energy of man has no meaning. Here, the potential energy of the earth-man system increases and also, work is done by the man while climbing stairs, irrespective of friction. The fact that work is totally (or partially) converted into some energy does not imply no work is done.
Part of your problem comes from thinking that the potential energy is somehow located in or a property of the person alone. And the way the subject is usually introduced could easily lead you to think that, but it's not right.
The potential energy is a property of the person-Earth system. In fact all potential energies are properties of systems of interacting bodies and can't be localized to any individual part of the system. We just find it useful to let the bit when gravitational potential energy involves the planet to go unsaid when we're working in a "near the surface of the Earth" context because the planet is playing it's part with every object all the time.
What this means is that
[...] the energy is converted to PE within system only. The person is the system.
isn't a valid way to think about this problem.
Work absolutely is done in moving the person up the stairs.
Firstly its the potential energy of the earth-object system as the two answers have said. "Potential energy of the object" is a loosely spoken phrase for the same for things happening on the surface of the earth where $g$ is taken to be a constant.
Secondly work is not done by a person. Work is done by a force.
There is an important thing you need to know. Consider climbing upstairs. Lets assume, for simplicity, that you can climb up just by pushing the horizontal surfaces vertically downwards (in reality friction will also come into play but don't worry thats not much important for what I am going to explain). Ideally you must lift yourself from rest in such a way that you do not accelerate i.e., the net force on you should always remain zero. Sounds strange? Indeed its strange because its a hypothetical process in order to define the gravitational potential. But thats how it is. You have to keep yourself in equilibrium while lifting so that the normal force exerted by the horizontal surfaces is always $mg$. You can imagine lifting your legs so very slowly to maintain this equilibrium every instant. This normal force does work and when you reach a height $h$ the work done is converted into potential energy $mgh$ increasing the potential energy (not the total energy) of the object(you)-earth system by $mgh$.
Two bodies can apply force on each other when they interact with each other like electric charges interact by their electric fields, masses by their gravitational fields, having physical contact between bodies etc.
The statement is false.
While the person climbs stairs, chemical reactions that occur in his body transform chemical energy into other forms of energy; PE and heat in the case under consideration. A part of the chemical energy expended appears in the form of increase in potential energy of the system (the person) and the remaining (in the simplest case) appears in the of heat that is dissipated into the surroundings. Thus the person expends more energy than increase in her/his PE.
Think of an inanimate object on a lower step. It cannot climb the stairs! Why? Because it cannot do any work!
There's no work done for a person climbing upstairs because the energy is converted to PE within system only. The person is the system. How true is the above statement?
I think it's true enough. You do work on a brick when you lift it up. You add energy to it, and we call this energy gravitational potential energy. Then when you drop the brick this potential energy is converted into kinetic energy which typically gets radiated away. Note the situation here is akin to what happens in explosions. Conservation of momentum means the Earth is also affected, but the energy is not shared equally, and we tend to ignore what happens to the Earth.
However when you lift yourself, all you're doing is converting chemical energy inside your body into this gravitational potential energy. You aren't adding any energy to your body, and whilst it can be hard work to climb up, and whilst you are exerting a force for a distance, you aren't doing any work on your body.
Does work done require interaction between system and surrounding?
I'd say yes. When the system is the brick, you're the surroundings, and you interact with it to lift it up and do work on it. But when you're the system there no energy being added.