Uno Platform is using the Android input behavior here, so when the input loses focus, the programmatic keyboard automatically disappears. However, if you want to show it again immediately, you can give it focus again from within the button Click handler.
XAML:
<TextBox x:Name="InputBox" />
<Button Click="TestClick">Test</Button>
C#:
private void TestClick(object sender, RoutedEventArgs e)
{
    InputBox.Focus(FocusState.Programmatic);
}
The input pane starts disappearing, but is immediately requested again, so it is almost imperceptible.
You can also try to trigger the soft keyboard programmatically without having any TextBox in place, by writing platform-specific code. This SO question for example includes quite a few solutions for Android. In general, they utilize the INPUT_METHOD_SERVICE and call ShowSoftInput on it.