I don’t see why I’m obligated to apply for asylum with the same passport that I used for entering the US. I’ve been living in the US on status for three years and my old passport is about to expire. I know that USCIS will know about my passport renewal before asylum. I just want to know if that’s an issue.
I'm glad that you are working out issues before applying. My advice is that you need to make sure that your facts are correct and line up. One assumption is that people granted asylum are 'stateless' = (they don't get benefits from their countries of origin. Benefits = including being issued a passport). Your facts have to line up: you can't claim being stateless, when you are still drawing benefits from your mother country, government issued benefits, including pension, passport, etc. Asylees/refugees, are people who have lost the protection of their country.
But then again, there is always exceptions. For example, if your asylum case is based on past torture or fear of future persecution by 'non-government' actors, theoretically, you can renew your passport, get your monthly pension check from your home country openly. But what you have to know is that 'situations' that you end up needing to explain to the judge/USCIS, diminish your chances of getting your case approved. The rule of thumb is simple: Do not do, or introduce variables to your case that make it hard to understand. Getting government benefits is one sure way to introduce a weird, difficult variable to your case.
Do not do anything additional that forces you to explain yourself to USCIS/Judge. Like, for most, we have been asylees for a long time. I suppose, we could all go to our home countries to take vacations, if it is now safe. Be we don't do that, we know we will have to explain these visits to the citizenship officer. Sure, one could say I went home because my grandmother died....but this could be evidence you are no longer a refugee. To avoid these situations, most of us are here, waiting for citizenship before going back, if it is now safe to do so. Good luck.