Putting a treadmill at an incline isn't the same as running up a hill

Most of those tests are such that exercise is just one factor out of many. I'd expect same results from either form if the amount/intensity are comparable. Given a similar heart rate you'll be consuming the same amount of oxygen, burning the same calories, and exercising your lungs the same amount.

One common test used is VO2MAX. This calculates the maximum amount of oxygen you can consume, divided by your weight. Test protocols vary but they typically involve using a treadmill and increasing speed and incline over time. If you wanted to max out this test you could prepare by running at an incline and at higher speeds. Shorter more intense sessions are more likely to increase VO2MAX.

This is however somewhat pointless. Most people (and a lot of what you find on the internet) makes wrong assumptions about VO2MAX. A typical bad assumption is that a person with a higher VO2Max will beat a person with a lower one at a sport. Current theory is that sports performance is not limited by ability to consume oxygen, but rather by feedback cycle designed to protect your body from damage.