The short answer is that yes, much of that is true (with some details being wrong, but the overall gist true). But none of it is because of your claiming the tax deduction; it's true regardless. In fact, you likely owe taxes for the caregiver as well - and certainly you'll need to pay those before claiming any tax deduction.
One important distinction is that you do not owe her a 1099; that would be for a contractor or a consultant. If she was your nanny for the summer, she was your employee, who is owed a W-2.
You should read over Publication 926, before you go any further. This goes over what your responsibilities are and what taxes you may owe. If you paid the student over $2,100 for the calendar year (this is the 2019 amount, check for your specific year if different), then you owe employer portions of taxes such as Medicare, Social Security, and FUTA (federal unemployment) taxes, and you should withhold the employee portions as well (or just pay them yourself). The employer plus employee portions of these taxes amount to over 20% of her total wages - before we get to income taxes (which you do not have to withhold).
The FUTA tax is also owed if you paid over $1,000 in a calendar quarter (meaning, April-June, July-September, etc.) to a household employee, so that's more likely, unless this is very occasional care (in which case it might not be qualified for the C&DC credit).
This also assumes she was over 18 - you did say college student, and most college students are; but if she was not over 18, then this doesn't apply in its entirety.
Publication 503 covers the Child and Dependent Care Expenses topic, including the tax credit. It doesn't explicitly say she has to have filed for taxes, though certainly you have to give her SSN and thus they know she didn't file taxes (though that might be okay, if she's under $12,200, the amount above which one is required to file). Publication 503 has a section for household employees, but it's pretty light on these kinds of details. It is clear though about the requirement to list her information:
You must identify all persons or organizations that provide care for your child or dependent. Use Form 2441, Part I, to show the information.
If you don't have any care providers and you are filing Form 2441 only to report taxable income in Part III, enter "none" in line 1, column (a).
Information needed. To identify the care provider, you must give the provider's:
Taxpayer identification number.
If the care provider is an individual, the taxpayer identification number is his or her social security number or individual taxpayer identification number. If the care provider is an organization, then it is the employer identification number (EIN).