I always encounter in my forays into the internet a certain way of talking and thinking about energy. The idea is that energy is a physical substance or field of forces that exists in itself like matter. Is energy really like this? Or is it a potential state like mass or motion that exists only relative to matter?
Energy, like mass, is a property of matter, not a substance in its own right. Energy and mass even have an equivalence, described by E = m c^2. Neither exist independently of matter, though.
So when statements are made about energy, like energy can't be destroyed, it just takes a different form, is there an erroneous assumption about energy that it is "in itself"?
Yes. The most common one is gravitational potential energy (say, a vase on a shelf) being converted briefly into kinetic energy (say, by a cat), and then into heat and sound. There is no energy "thing"; it's really just an accounting system for different properties and states. BTW, energy isn't only a property of matter. Photons have energy too. And they can impart their energy to matter.
No, it's not really like that at all. That's a very common misconception. Energy, ultimately, is just a number (with some units). It's a useful concept because it turns out that, in many processes, if we calculate the total value of the number at the start of some process and then again at the end of the process, we find that the total is the same before and after, even though the "system" under consideration has undergone various changes. We can use the concept of energy conservation as one more (very useful) method for making predictions about how systems will behave. All of the following statements are essentially meaningless: "Mass is a form of energy" "Matter is just concentrated energy" "Everybody has a soul that is made of 'pure energy'" "When matter and antimatter come together, they annihilate to produce 'pure energy'" "God is a being of 'pure energy'" "Everybody has an 'energy aura' that surrounds them" "E=mc^2 tells us that mass and energy are equivalent" "Ghosts are beings that are 'pure energy'" "Sunlight is a form of energy" "Energy comes to my house through the electricity wires" "My car runs on energy"
Might we better think of energy as a mathematical property instead of a physical property? Also, how does light have energy if it has no mass?
Linear and angular momentum are similar to energy. They too are conserved quantities that are a bit abstract, in that they result from multiplying various physical quantities together. But when you think about it, almost any physical quantity is measured indirectly. Mass can be measured through weighing things, but what one observes is a deflection of a pointer or something that depends on a chain of relationships to convert it to mass. Same with velocity. One observes two positions of an object and uses a timer, then calculates what this thing called "velocity" must be. So I don't think energy is any less physical - or more mathematical - than these other quantities.
