My son exhibited similar behaviors around the same age. I think similar questions on this site show that many parents have this problem, so you're definitely not alone here.
However, knowing other people have the same problem doesn't make this any less frustrating!
My wife found it slightly annoying, being followed or cried after whenever she moved around. I was probably more frustrated by the fact that my son was interested in me once my wife left the room. So, this situation can be emotionally difficult for both parents.
What got us through this was just understanding that it's a common phase, and eventually he'd grow out of it. Part of it was likely due to me already being gone from work and school so much that my wife was almost always there.
It's also not unusual that they behave differently at day care, because it's a completely different environment for them.
This Ask Dr. Sears article addresses how to raise an independent toddler, and I think some of the solutions here may be useful for you. And, it sounds like you're already doing some of the things it mentions:
- Keeping a toddler posted on your absence
- Substituting voice contact
So I would try some more, and keep up with these.
For reference, my wife and I still keep my son posted on what we're doing, and talk to him even when he's acting out about us being gone. The most notable example of this is when we're using the bathroom. Both of us tend to tell him that we're going potty. This helps him learn that such absences (potty) are short, we can still talk through the door, and (we hope) it may help facilitate future potty-training. We also tell him whenever one of us is leaving, such as "Papa is going to his room." or "Papa is going to work." or "Mama is taking out the garbage and she'll be right back."
Lastly, it did also help when there were some times when it was just me and my boy, and my wife was out on her own. He would sometimes fuss for awhile because she wasn't there, but eventually he realized that "Mama is gone" time pretty much meant "Papa is playing" time, and her absences began to indicate fun instead of stress.