“ Which do I need to file with my return? Do I need to claim both?”---> I believe that your spouse’s 1099 money gets taxed as ordinary income by the IRS. So, she'd just add it to her income on W2 for the year. Add it on top of her wages and put it in the wages box; however, if the money on 1099-MISC is for damages your spouse suffered and the payment is for money your spouse lost then it would not be taxable. She gets a 1099 for the money paid to her (outside of your job) over $599( depending on the type of 1099 and the nature of the payment, the minimum amount varies considerably.) for IRS and the amount varies from state to state.
“And why was she issued both forms if it was taxable income?”---->The amount on the 1099-MISC is money paid to your spouse outside of her job. A 1099-MISC is a type of tax form. It is used to report miscellaneous income, such as income earned as a non-employee.