That’s not the claim they’re making, they’re saying wind is cheaper than just buying the fuel for a pre-existing natural gas plant.

I don't understand how you can compare the cost of a wind farm to fossil fuel. The former provides megawatts, and the latter provides megawatt-hours.

You would have to incorporate assumptions on lifespan, maintenance and/or amortization.

Yes, that’s what they do, it’s called a levelized cost. These costs usually come with a sensitivity analysis of the impact of the different assumptions.

Wind or solar is actually often commissioned using a power purchase agreement for a certain price per kWh, guaranteed over a certain period.

Unless the wind turbines are trapped in some sort of time freezing status field, I would generally assume that they both can be measured in megawatt-hours. Or you can multiply out by 3.6 to get to gigajoules if you want more sensible units to play with.