Running apps don't show up in dock

### What Happened?

Certain running applications never show in the dock.

- Default apps (the ones shipped with the OS) show up fine
- Apps downloaded and installed from other sources (e. g. `apt`) do not. One exception I found is Google Chrome.

### Steps to Reproduce

Example 1:
1. `dpkg -i docker-desktop.deb`
2. Click Applications menu, run Docker Desktop
3. Docker Desktop is launched
4. It is not visible in dock

Example 2:
1. Add a new .desktop entry to _usr/share/applications_, for example:
```
[Desktop Entry]
Name=IntelliJ IDEA
Exec=/path/to/idea-IU-252.27397.103/bin/idea
StartupNotify=true
Terminal=false
Icon=/path/to/idea-IU-252.27397.103/bin/idea.png
```
2. Go to Applications menu and run IntelliJ IDEA
3. IDEA is launched
3. It is not shown in the dock

Example 3:
1. Drag the IDEA icon from the Applications menu to dock. It is now pinned to the dock.
2. Launch it from the dock
3. The application launches but the white dot (indicating running app) is not shown under the icon in dock.
4. Launch it from the Applications menu
5. Same as 3.

### Expected Behavior

App is shown in dock. It used to work in previous versions (I haven't used Elementary in a while, I remember using versions 6 and 7 in the past. This is a fresh install of 8.

### OS Version

8.x (Circe)

### OS Architecture

amd64 (on most hardwares)

### Session Type

Classic Session (X11, This is the default)

### Software Version

Latest release (I have run all updates)

### Log Output

```shell

```

### Hardware Info

_No response_