I think he does - but he needs to “be” Clark to love her like that. When he’s Superman he can’t think like that, because his base nature is to think holistically and perfectly, and there’s no room to focus just on Lois.
I think Lois probably understands that too – he’s an alien so there’s a part of him that she can never ever really know, and that’s the “Superman” part (but as Mark and Maggin said, that’s also the part of him she loves).
I thought it was interesting thinking about the three-faces take in the context of Batman because it’s kind of the reverse. Bruce Wayne comes first and he adopts the Batman identity for the purpose of terrorizing criminals (as opposed to Superman adopting the Clark identity for the purpose of becoming human), but Bruce also has a false public identity which exists to ridicule any investigators who might make a connection between the two, and his real identity is the Bruce Wayne we see in the cave when he’s with Alfred. Superman on the other hand has the fortress, but no Alfred, so he’s far more alone, and has a far bigger responsibility than Bruce.