First it does require that you specify a library. However when it takes you to the sign-in screen there is an option to sign in with your Overdrive account instead of your library card for that particular library.
I believe Kobos do it this way because they can only display a single library's results. It's kind of like how in Libby, even if you have multiple library cards, you have to choose which one to use before making a search. The UI will not search across all your cards and combine the results for you. Likewise the Kobo still has to choose a particular library to search – but if you're logged into your Overdrive account, then you should still automatically receive your loans made from other libraries on your same Overdrive account.
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