Owed tax on 50k is roughly zero after the child tax credit (note: you will actually owe some if you got those advance child tax credit checks in the mail) That’s why they’re telling her no withholding is needed. You didn’t put both kids (or have them put that for you, however that worked) so you’re getting the small withholding. That’s all fine so far.
What’s more important though: the second income is all tax, and all at a higher rate for the most part, since it’s on top of the first one. So it may need more withholding.
When you have two incomes, there’s a box you check ‘withhold at higher single rate’. Do that on one of your two W4s - doesn’t matter which if they’re both the same amount. That way one of your checks withholds the higher amount. I also would put zero dependents on that one. That way you aren’t being double counted.
The problem is that the taxes are like this:
50k: -12950 (personal deduction) = taxable 37050. Tax on the first 10k or so is 10%, and the next $30k is 12%. That’s nearly $4k- and since you have two kids the child tax credit is $4k, which means owed tax is zero. Great!
But add the second 50k:
50k-12950 = 37050. Around 3K of that is taxed at 12%… the rest is taxed at 22%. So a bit over $5k of tax for this portion, and no child tax credit to offset it. That’s as opposed to under $4k for hers - over $1k more for the second income even before the credit.
It might be less based on the fact that you have health insurance and such probably coming out. That’s all effectively coming out of the 22% part. But point is, it’s more. Check that box on yours and it will be correct.