I have been on both sides of this, both feeling like I "should" hold back when our oldest was tiny, especially when she was very young, fearing that I might "mess up" nursing if I pushed for too much time/contact with her, and on the other side, after I gave birth to our youngest, and was frustrated that my wife seemed to be hanging back, prioritizing everything else in life (and admittedly, there was a LOT) above our new baby.
I'll echo the suggestions of time alone, both long stretches during the day (we set up leaves to make sure this happened for the non-bio-mom both times), and being out *in public* as osker smartly suggests. Despite being pretty aware of the issues, and very communicative, I didn't really gain my confidence as a mom until allthe baby decisions were on my watch for pretty long stretches of time, when I was primarily home with her from months two to 6. It definitely felt like a shift from our first two months both being home. I was forced to notice and decide when she was hungry, tired, etc. I couldn't defer.
If your partner is doing any feedings, I definitely recommend doing skin-to-skin during bottle feedings and making them long and snuggly.
One thing I also wonder is how much your partner is talking to you about how she's feeling, both during pregnancy and now? Are you just picking up that she's struggling or is she really talking about it? Did you get a sense that she struggled at all during pregnancy? I am an *extremely* communicative and forthright person, pretty darn in touch with most of what goes on in my head and my heart, and DAMN was I confused during that time. I really felt like I was flying blind, and thus that *we* were flying blind, and that was with an extremely supportive partner who (mostly) took my concerns, ideas and worries at face value (I.e. she didn't tell me I was "making things up" -- not that you are! But it happens). All of this to say if she's giving you *any* hints of what's going on for her, jump on it, really listen and dig in. It is *hard* to know how to talk about it, hard to even find words for the feelings and thoughts, because there is no script. The more you can draw her out on it, the more you can both engage and she can really feel your support, the better (at least that's what worked in our house).
Do consider looking up our blog. We've written so much about this stuff. Look up "first time second time" and in particular look for the post called "choosing to parent" (there's a search bar on the side, it's from July 2009, back when our second was a baby). It describes my wife's thoughts in shoes very similar to your partners, why she was taking a backseat, and what she did to get on more solid footing. And seriously, we have written tons on non-bio/bio-mom dynamics as they've played out at our house, so you both might be able to find something to relate to in there (we write about it precisely b/c it can be kind of hard to find support for situations like this. Even though we all go through them, a lot of it feels kind of like stuff you aren't supposed to talk about out loud).
Hang in there and good luck to all three of you.