Neither....
Hey! I always assumed that upon copying the DNA of an animal into yourself, you are LITERALLY COPYING EVERYTHING about that animal that is stored within its DNA. Like how it looks at the equivalent age (characteristics), how tall and heavy it is (mass), its senses and capabilities, its instincts (mind) and all.
So its physical AND the mental 'copy' would literally be made, 'copied' into an exact replica and 'stored' in Z-Space.
And the morphing energy is the power that changes each cell on the morpher's body into that of the animal it is going to be come.
All very abstract, I know.
But the morpher is able to focus on the animal to initiate change and thus the morphing begins. The transfer of the excess mass would be from the 'made' copy and not 'borrowed' from some other source. So I would imagine, if I was to morph a bug, ALL MY ATOMS would MORPH into that of said bug, and as my mind stays in 3-D space along with the bug body and the bug's mind, my physical form would be literally floating comatose in Z-Space nowheres, until my mind in the 3-D plane would call upon it and demorph back.
I'd like to add the topic here, that I never quite got the mechanics of the Escafil device. Supposedly a morph-capable person with knowledge of the tech can transfer the tech. But then came the ant which happened to touch the blue cube and then touched Cassie, it being a non-sentient creature with barely any self-awareness in the series, MANAGED to acquire Cassie and morph into a human.
How the heck did that happen????????!
And in book #1 (I know, I know there are a lot of K.A.S.U.s based on Michael and Katherine still establishing 'the ropes' of the whole Animorphs philosophy) Jake as Homer, smells all sorts of things and identifies dogs from down the road... the authors were trying to play with the concept of memory and didn't quite follow it up, like they were undecided after that.
But by the time Cassie morphed Rachel in #12: The Reaction she implies something similar again. How Cassie says Rachel's brain just wants to do stupid things...
We know that instincts are stored in DNA, but could the morphing process in the series suggest that things like 'memory' and 'personality of sentient species' are also stored in DNA?! If not, this is certainly implied through the Arns' ixcilla.
So I imagine under this technology of morphing comes the components of DNA, cell transformation, inter-dimensional transference (whoa!), healing (bc the process doesn't hurt nor retain injuries), telepathy and some extra component that enables the morpher to bring in clothing, and their water and food and stuff like the Yeerk pest inside their head with them.