It seems Java supports showing operating system native splash screen while the app is being started.

I only tested it on Windows and it worked as expected.

So, in my project build.gradle(.kts) I added this:

compose.desktop {
    application {
        nativeDistributions {
            appResourcesRootDir = rootDir.resolve("myAssets")
            // ...
        }
        // The image path is relative to the app (Java) working directory
        // which is the app installation root directory (where the .exe file resides)
        jvmArgs += "-splash:app/resources/myImage.gif"
        // To make it work whatever is the working directory when invoking the app use:
        // jvmArgs += "-splash:${'$'}APPDIR/resources/splash.png"
        // ...
    }
}

It uses the myImage.gif image placed in <PROJECT_DIR>/myAssets/common/ directory.
See Adding files to packaged applications for more information.

The splash screen works when the app is launched with its installed exe or one of Compose Multiplatform run*Distributable tasks.
To also set the splash for application fat/uber jar (created with one of package*UberJar* tasks),
copy the splash image into classpath (like src/main/resources directory)
and add the code below in your build file to update all the Jar tasks:

tasks.withType<org.gradle.jvm.tasks.Jar> {
    manifest {
        attributes["SplashScreen-Image"] = "myImage.gif"
    }
}

It seems that Windows does not respect the GIF speed and no-loop settings and
plays the animation at a low frame rate. See Gif Animated Files in C# have Lower framerates than they should.

See the related issue in Compose Multiplatform repository.