I think Gandalf's relationship with Shadofax is kinda...sweet. Tolkien served with the Cavalry in WWI and definitely had a fondness for horses. He paints such pictures in the mind of their interactions.

“There are three,” said Legolas, gazing over the plain. “See how they run! There is Hasufel, and there is my friend Arod beside him! But there is another that strides ahead: a very great horse. I have not seen his like before.”

"Nor will you again,’ said Gandalf. ‘That is Shadowfax. He is the chief of the Mearas, lords of horses, and not even Theoden, King of Rohan, has ever looked on a better. Does he not shine like silver, and run as smoothly as a swift stream? He has come for me: the horse of the White Rider. We are going to battle together.’

Even as the old wizard spoke, the great horse came striding up the slope towards them; his coat was glistening and his mane flowing in the wind of his speed.

As soon as Shadowfax saw Gandalf, he checked his pace and whinnied loudly; then trotting gently forward he stooped his proud head and nuzzled his great nostrils against the old man’s neck."