How to style the standard react-native android picker?

It can be styled via native android. See this and this.

Add the following code to /res/values/styles.xml

<style name="AppTheme" parent="Theme.AppCompat.Light.NoActionBar">
  <item name="android:spinnerItemStyle">@style/SpinnerItem</item>
  <item name="android:spinnerDropDownItemStyle">@style/SpinnerDropDownItem</item>
</style>

<style name="SpinnerItem" parent="Theme.AppCompat.Light.NoActionBar">
  <item name="android:fontFamily">sans-serif-light</item>
  <item name="android:textSize">18dp</item>
</style>

<style name="SpinnerDropDownItem" parent="Theme.AppCompat.Light.NoActionBar">
    <item name="android:textColor">#ffffff</item>
    <item name="android:textSize">18dp</item>
    <item name="android:fontFamily">sans-serif-light</item>
    <item name="android:gravity">center</item>
    <item name="android:background">@drawable/mydivider</item>
</style>

Create a file at res/drawable/mydivider.xml and add the following code

<shape xmlns:android="http://schemas.android.com/apk/res/android"
    android:shape="rectangle">
    <solid android:color="#29A1C9" />
    <corners android:radius="0.5dp" />
    <stroke
        android:color="#FFFFFF"
        android:width="0.1dp" />
</shape>

Before styling:

After styling: