"Is is possible . . ?"

If you define riches in terms of possesions which improve your quality of life, yes.
If you define riches in terms of possesions held in relation by yourself against possensions not held by yourself, then no.

If two people living in different climates successfully share their houses as holiday homes with the other, have they not both become richer?

That is an example of a simple principle. All involved becomes richer through the increased efficiency of any usage of available resources.

Likewise non-physical riches, such as digital copies of music, can be possessed by any number of people.

Through increased technology we are also granted riches which did not even exist previously.

In answer to your question. Yes, we can become richer without others losing out. However there are limits to how much and in what ways we can do so.

We can't become richer without others losing out. No matter what. But that is not to say that if you make a million someone will starve. That million wouldn't have ended up somewhere else were you not to make it. But you made it, so the other guy who was trying didn't. He's worse off, but now I think you'll agree that that's not your problem. Competition is key.

the finite resources of this planet must be shared equitably for all to 'succeed'; not to mention the entire system of sharing needs to be overhauled.

['ownership' is based on 'faith', and all such things are vestiges of a distant, now meaningless, primitive past.]

Yes, you can create riches without making someone else worse off, by producing something new using your own labor and resources and by fair trade.

The problem is that it is easier to acquire wealth by under-compensating other people for their labor and property under a system that allows a few people to monopolize resources (land, hunting/fishing rights, minerals, taxes/tithes) for their own benefit.

If resources are limited, one person's over-use of them seems inconsequential but eventually deprives everyone (the tragedy of the Commons) such as deforestation, overfishing, hunting species to extinction, and pollution.