If you can limit the number of registered nodes per person, do you think such a system will become very fair?
No, it will increase pooled mining. Currently, you can mine coins without running a full node. If you restrict the number of nodes, then there will be even less miners with their own node. You will only harm people that run nodes without mining, which is probably not what you want.
The Pow mechanism is very complete and mature, but its node power is concentrated in the hands of wealthy people.
Not really, because anyone can run a non-mining node. And it seems to be useful in some situations, for example here:
https://bitcointalksearch.org/topic/error-connectblock-too-many-sigops-invalidchainfound-invalid-block-5447129Without full node running 24/7, getting block 00000000000000000002ec935e245f8ae70fc68cc828f05bf4cfa002668599e4 would be very hard, because block explorers won't show you that, so you won't see, why this block was not accepted.
In theory, nothing stops non-mining nodes from charging fees for their services. In practice, most of them are not rewarded in any way, the only reward is that the node owner can be 100% sure what is going on in the network, if that node is online 24/7.
Mining pools are a large node. Will such centralized computing power affect the security of Bitcoin?
Why you want to solve "large node" problem by limiting the number of nodes? You should do it in exactly opposite way: encourage people to run nodes, so there will be a lot of smaller ones.