"Can an emission analyzer (for automotive exhaust gas) be useful if my main concern is the oil inside the engine?"
Measuring the health of oil inside an engine from automotive exhaust gases would be extremely difficult if not impossible. This is because oil should not be present in the exhaust gas regions. Even if the technology was available to accurately measure and analyze tiny particles of oil in exhaust gases, the oil would need to pass through the exhaust environment with carbon/soot and the superheated region of the catalytic converter, which would essentially turn the oil particles into carbon dust. Therefore, the likelihood of obtaining usable data would be non-existent.
The questions you should be asking about your engine oil require a carefully collected sample at a significant volume, both of which would be unachievable with an emission analyzer. One of the best ways to capture data from your engine oil is to take a sample directly from the sump using a sample pump. This will allow you to obtain the volume of oil needed to determine the health of the engine oil.
Among the types of data you will want to capture from the oil is the viscosity, which can be a great indicator of the oil's health. When viscosity starts to rise, the oil is approaching the end of its lifespan. The antioxidants are being used up and have been enlarged by the pro-oxidants (soot). This is the sludge you see in the bottom of your engine's sump.
Another important indicator of oil health is the acid number. When the acid number goes up, the additives in the oil are being depleted, and the life of the oil is coming to an end. This also applies for the oil's base number.
The oil's particle count can also provide a lot of good information, such as whether there is any wear debris (metal/bearing wear) or contamination of the oil from dust ingression through the air filter.
In conclusion, there is no way to properly measure the health of your engine oil through the exhaust gases. The only appropriate method for testing the oil would be through conventional oil analysis, as described above. Of course, there are also some simple tests that you can perform, like a crackle test or a patch test, if you would like to test the oil yourself.