It's currently not possible to use the full version of Spotify via the web player on iPad: if you go to open.spotify.com, you can only listen to 30-second previews of songs, and there's no option to log in (note that this doesn't just apply to Safari).
If you'd like for this feature to be implemented, make sure to head over here and create a new idea. That way, other users can show their support for the idea, and the more people that vote, the more likely it is to be implemented by Spotify in the future.
Thanks! Stay awesome 🙂