I'm not in the USA so I'm not sure how the W2s and SSNs are tied to job/healthcare/401k etc, but how does this double timing work? Won't the Tax office/Finanzamt (IRS) know about this?
They can also request that the employer withhold additional taxes; this is fairly common because of situations where both spouses work.
They likely have two healthcare plans, but that's generally not an issue - again, it's fairly common if both spouses are employed.
For the 401K, the individual will just have to make sure that their total contributions between both companies are under the federal limit.
The IRS knows but they don't tell employers.