Can you use one Stream Deck with the Elgato app and one with SoundFlow's?

Yes and no.

You can only use one app to control your Stream Decks. But if you choose to use Elgato's Stream Deck app, you can use the SoundFlow plugin to let SoundFlow connect via the Elgato app, and so effectively have Elgato's Stream Deck app be "in charge" and SoundFlow can then operate as a guest.
You can view the guide on how to set this up on the following link, if you scroll past the "3.7" guide down to the old guide:
https://forum.soundflow.org/-1849/how-to-set-up-soundflow-stream-deck

Note that we don't recommend this as it's harder to set up correctly, it's slower, and you're losing a lot of the niceness of having your full setup be saved in just one app (SF). When you design all your Decks in SoundFlow, you can move to any other computer and just log in and have all of your settings available. This also means they're backed up every day for you, it means you can share your Decks on the Store with other users, and vice-versa install Decks from others.
Anything you design in the Elgato app is only available there, and if you move computers, you'll have to manually copy files from various different directories.

Typically people ask this question because they believe there's something they can only do in the Stream Deck app and that isn't available in SoundFlow. Increasingly though, we do support most scenarios directly in SoundFlow.

So my follow up question would be - why do you want to set it up that way? I bet there's a better solution that lets you keep more of the benefits of SoundFlow, depending on what you're trying to do.