I can't speak to how Turbotax wants you to answer this question, I know there is a lot of confusion.
Under the law, what counts is that your spouse is covered by an HDHP, even if the policy is not in her name. HSA'S are like bank accounts except they are owned by one individual, there are no joint accounts. Even though "we" contribute, it is only "your" account. Your wife could open an HSA in her own name if she wanted to, there are many banks that offer this, it does not have to be through an employer. And she can make tax-deductible contributions in her own name, as long as she is covered by an HDHP and has no "other" medical coverage. Because you are covered by a family HDHP, your maximum contribution for 2023 is $7750 and for 2024 is $8300. You can split that any way you like. If one or both of you is age 55 or older, you get an additional $1000 catchup contribution, but that only be contributed to each individual account.
