I thought this was an easy question and it is if you believe that the world consists of just two, equities and fixed income; however, the complexities mount and you try and break the world into finer categories. You will always find that most questions are easy until you have to provide your own useful working answer.
To define an asset class, you have to ask, what is the essence of the class? The essence of any object is its attributes or its substance. The essence is what it has to be its necessity. What does the object do? If it loses those special attributes that are necessary to its existence, it loses its identity. In philosophy, we can go back to Aristotle and discuss definition or essence through the word, horismos as described in the Metaphysics.
So why go back to philosophy to discuss the definition of asset classes? The answer is simple. Describing or using the right definition is not easy and you want to be more thoughtful than just providing the obvious. An equity is the residual value of the firm and the discounted futures earnings of the firm. Its essence is associated with risky cash flows. A bond is a claim on the firm and the discounted cash flows associated with a coupon. We all know the difference, but sometimes the essence of discounted cash flows become harder to describe when both have risks. For example, what about a credit? Credit products are not the same as a government bond, but it also is not like discounted earnings. There are risky cash flows, but a credit is somewhere in between. Hence, it has characteristics of both. The risks of some assets are not so simple to describe. The cash flows of any assets have different levels or sensitivities to risk.
You can also think of an asset class through how it will behave to different risk factors. What is the sensitivity of equities to a change in inflation? The sensitivity will be different than what you will see with a bond. Is the essence of an asset class its sensitivity to a particular risk factor? You can classify an an asset by the type of risk premium that it is associated with. If the essence is related to its sensitivity to risk factors, does that mean an asset could move between classes based on how its sensitivity to risks change? You can have equities that behave very differently because their risk sensitivities are different.
Some may not want to provide any definition but use a clustering methodology to describe or bundle assets into a class. This means there could be degrees of equity-ness based on where an asset fits within a cluster. Makes sense from an empirical point of view but what happens if the statistics change?
A problem could exist with even geographical representations. Is a multi-national corporation with worldwide sales a US company? A UK headquartered firm may have little in common with local business or the country economy.
Before you throw-up your hands, I will state that categories can be made and asset can be placed in these definitional boxes, but there are no hard and fast rules. We can define an asset by it cash flow types, but we also want to say that the essence of an asset is its sensitivity to risk factors. We ultimately bundle risk together in a portfolio so there is no getting away from risk sensitivities regardless of how you define an asset.
So we make classifications and try our best, but accept that an asset class is not stable but a dynamic concept.