When we were homeowners, we definitely counted the house as part of our Net Worth, though how much of a true "asset" it is probably depends on how leveraged you are with a mortgage, and current market conditions. Until we were confident, based on comps, that we were in the money on a sale, we didn't considered our home a liability/debt.
We're renting now, but plan on buying again when we FIRE. At that point, our plan is to consider the house part/all of our REIT allocation and rebalance our AA accordingly. That said, whether we consider the home an asset will depend on what we pay, the mortgage we get, and market conditions.