The EIN is a nine digit number usually 2 digits with a dash and 7 more digits. If other taxpayers have filed using the state license number, then they may not be using TurboTax, or the IRS has not caught up with them yet.
You are required to use either the daycare provider's EIN, if they are a business, or their Social Security number, if they are an individual. The daycare provider is required by law to provide you with their tax number, with a $50 fine if they do not.
Download and print out form W-10 from the IRS website and give it to the daycare provider. If they refuse to provide the tax number, then you cannot e-file your tax return. You can print your return and mail it in. Write "refused" in the space for the daycare provider's tax number, and attach a separate written statement that gives the providers name, address, other identification, and describes the efforts that you made to obtain the tax number.
I suppose there is a chance that the state license number is the same as the federal tax number, but if TurboTax is rejecting the number because it is in the wrong format or is invalid, then the state license number is probably not the same as the federal tax number.