Our children were terrible sleepers. Our eldest did OK at night and usually only got up 3 times a night from the time he was little, and he went back to sleep easily, but he continued to wake up once a night for a long time (until he was probably 3 years old). He napped like shit, though. I love that kid, but I would have mental break downs because he wouldn't nap unless I was holding him. That might not sound bad to many people, but he was a baby that needed to be held literally all the time, or else he'd scream bloody murder. He didn't nap in a crib or anything until he was closer to a year old. He would fall asleep in my arms, and I'd slowly, slowly transfer him to somewhere else (a soft pad or an infant seat or something), and he'd wake up 10-15 minutes later screaming because I wasn't holding him. I couldn't shower, couldn't make dinner, couldn't do dishes, etc. I felt so much love for that little guy, but I also felt so much anger during those moments. I knew that wasn't healthy, and so there were times I would have to put him in the crib and walk away and feel like the worst parent alive while he screamed bloody murder from his room. So many times I called my husband in tears...
Our daughter was the opposite. She was a dream baby during the day. It was like a gift from the gods! ...until the sun set. She would not sleep at night. I started to quite literally feel like I was going insane.
I would like to be able to tell you there is a simple fix for these things. There isn't. The sleep deprivation is so bad. It's the reason my husband and I aren't having any more kids ever. We love our kids. We love being parents. We are those sickeningly sweet people who are just so happy to have children. They're fun, they're hilarious, they're cute. If they started out happy and healthy at 1 year old, I think we'd have a dozen kids. But that first year is the stuff of nightmares. We knew if we had more kids we wouldn't survive it. Our marriage would break apart or something. Who knows? But we couldn't do it again.
What you might want to consider is each getting 1 weekend night to sleep uninterrupted. I know she's breastfeeding, so she might need to be woken up once or twice to nurse, but on her nights to sleep you can bring the baby directly to her in bed and then whisk the baby away when he's done eating. Maybe she can have a bottle pumped so that if he wakes up at 5am or something, you can feed him so she can sleep in a little. This way, you each can look forward to 1 night to catch up on some sleep. Another option might be to each take half of the night shift.