I don't lie; especially in my profession. It's a thing. I know that personal Integrity is considered a "quaint anachronism," in today's world, but I won't compromise on that.
You’re right not to compromise and even if you were less ethical, that’s something which is easily detected and could lead to being fired with no recourse. It’s very reasonable for a company to say someone who lies about simple facts can’t be trusted with anything else.
You certainly don't have to lie, but there's no reason you can't simply calculate your age differently. The common 365 days to a year for age isn't a universal given.