Spotify is gearing up to add track rearrangement to playlists on the Android app at some point in the future. Though it's not currently possible to change the order of tracks in playlists on Android, the Spotify Community Team has mentioned in a comment to a submitted post on the community forum that it is working on getting the feature added. Spotify will start with small batch rollout of the track rearrangement at first, but will eventually put it out there for everyone to enjoy. Starting things off with a smaller number of users may allow Spotify to basically test this feature out, so it shouldn't be too surprising that it will hit Android in phases as this often happens with app updates.
Spotify also says this will begin hitting a small number of users in an upcoming version of the Android app, so it will be part of an update and not a feature that Spotify simply has to turn on server side, which means users should be able to better recognize that the feature will be available as Spotify will also likely list it in the changelog.
Rearranging tracks in a playlist is available on other platforms. For example the iOS app version of Spotify allows users to put songs in any order they like. You can also rearrange tracks for playlists with the Spotify desktop app. Worth noting though is that you can only rearrange the tracks for playlists you have personally created, and not pre-configured playlists. When Spotify finally adds this feature to the Android version of the app it's highly likely that this particular trait will come along for the ride. Spotify says they'll end up putting out another post once they have more to share about the feature coming to Android, so it may end up alerting users to the feature's release date before it actually gets pushed out.