Looks like your web map was configured in the "new" map viewer. Web Apps built with WAB don't support that and they're still based on map viewer classic. If you build the app in Experience Builder you should see the pop-up as its configured in the web map!
Experience Builder has a lot of pre-made templates that are based off of WAB layouts, so hopefully that will help with the transition!
Best,
Katie
“The goal is not simply to ‘work hard, play hard.’ The goal is to make our work and our play indistinguishable.”
- Simon Sinek