Consciousness may not be possible without time, but how are consciousness and identity bound up together? For instance, if I get amnesia, I have a single consciousness and multiple identities.
Kant would assert identity outside time even more completely, as he was a Christian from a tradition of personalizing God, but leaving him 'eternal' in the Augustinian sense. (I am not suggesting you consider Christian dogma as philosophy. But think about how those before you thought about time through the prism of Augustine's notion of a God outside time.)
From that point of view God should be able to have an identity, but he should not be limited by time. It is not true that he could not have consciousness (the Christian claim (and the Hindu one) after all, is that he incarnated himself, perhaps, a'la C.S. Lewis, for the very purpose of experiencing consciousness.) But as an omniscient, changeless entity, fully aware of the future, he would have no need of consciousness, knowing everything, and having come to know everything 'all at once', he would have no need to learn or to have experience. So identity and time need not be related, if you have a notion of thought and will that is 'perfect and unchanging' and not sequential.
So time is only related to the one factor. Time and consciousness are related. From Kant's point of view, that relationship is almost complete, and it is not shaped the way we might first assume. It is time that is an aspect of consciousness, and not the other way around. Time cannot ultimately be an underlying principle, like existence, it has to be a contingent way of our interpreting our own experience for the sake of our limited minds. So having presumed consciousness, it would not be necessary to separately presume time.
The modern notion of 'spacetime' from Einstein makes this more credible, as we know that each individual travels through time in a way that makes it slightly different, according to what speeds and other changes are involved. For there to be a single, underlying structure there, it needs to be either very complex or quite odd in some other way we have a hard time grasping. Whereas if only the composite of space and time in experience exists, and neither space nor time are absolutely real on their own, what are the odds that either of them, or both of them together, are independent of us, and do not proceed from our interpretation?