There’s no doubt that a CS major will be better for a software engineer position, but what about a mathematics major? Will this make me more unique from other applicants?
I’m not sure if you really want to be “unique” so much as “better qualified.”
Additional math courses may be useful in some areas (e.g. algebra and number theory for cryptography), but generally CS is more directly applicable to software jobs than math. Math graduates are more likely to be able to self-educate the needed CS later (and may have been more likely to take some CS courses as electives in college) than many majors, but if the actual goal is a software job, majoring in math is a rather roundabout way compared to majoring in CS.
Now, if your true interest is math, and a software job is a backup option behind a more math-based option (e.g. -> PhD -> academic research in math), then you may major in math while taking some CS courses as electives to prepare for the backup option if needed.