To your first question, no, this is not odd. But it is a sad situation...
I was a nanny for years, myself. Many of us who have studied child development *know* that kids need hugs and caring touch to grow. Some kids are more affectionate than others, just as some adults are. Kids who want affection, who feel comfortably bonded and happy and trusting with their caregivers will freely seek this, and will continue to express connection and affection if the caregiver reciprocates, giving the 'green light' to the child that their affection is welcome.
When my son had a caregiver ( a neighbor, stay at home dad) while I was working, he got loved up a lot by this other father. I really enjoyed that he had other people in his life who he was close to, whom he felt he could depend on and trust. They are still close, and I appreciate that relationship so much. So does my husband.
Your husband has to set the precedent with his own kids and let them know he is open to affection. If they have been getting a red or yellow light (so to speak) about hugging and kissing dad, it's his job to start demonstrating 'green light' behavior. That means that he's going to have to be assertive in giving them love and affection in ways that feel comfortable for him. Even if you fire the nanny, *you will still have the same problem* with the next one, male or female.
Would your husband be willing to examine his own feelings and possibly go to talk to someone about *how* to bridge this gap? I think that would be a great first step. Firing the nanny will only create distress and loss for the children who have bonded to him. Very good, loving and responsible nannies aren't the easiest people to find. I truly hope your husband can work through this, because I'm sure it's hard for him, too. Good luck!
Added: after reading a comment or two, I just want to confirm what many of the more open-minded moms have suggested: male caregivers can be wonderful for children. I've worked with several great guys in my line of work and it's often said among other female caregivers: we need more men in this line of work. Men bring a great sensibility and many children need to experience caring, involved males. I've encountered *far more* unstable/emotionally unsafe women in this business than I have men (only ONE).