How to log an entity that has collections?
Currently no bundles/extensions have this functionality out of the box. One option would be to implement it yourself. This can be done by making use of Doctrine Listeners. Particularly you need to listen to postUpdate and postPersist events - these happen when entity is updated and created and store your Tags there.
Another option is to get rid of ManyToMany relationship. For this create an intermediate entity AssetTag that would have OneToMany relationship to both Asset and Tag. After this is done, you can use EntityAudit Doctrine Extension, which supports this type of relationships.