1. If you daven in a small shul, it is unlikely there will be more than two people with the same yertzeit (unless the parent was killed in a mass casulty incident, though except for 9-11 there really haven’t been any since World War II). The bigger the shul, the more likely there are multiple people with yerzeits. Presumably you are a “regular” in the shul. If you are able/expected to make kiddush the Shabbos before, and to bring food on the day of the yertziet, that smooths things over.
2. There is no halacha you have to daven all tefilos in their entirely on a yertzeit. Sharing is customary. Obviously this is a problem if you follow the western European “yekke” minhag that only one person says kaddish at a time — due to many mass casulty incidents in the past, most Asheknazim adopted the minhag of everyone with a hi’yuv say kaddish together.
3. If you regularly learn, why would learning on the yerzeit be a problem. If you don’t regularly learn Torah, that is a much bigger problem.