We know Saber's real identity is Arthur Pendragon ("Chief-Dragon") and that she is linked to dragons through Merlin's sorcery:
King Arthur was made into the incarnation of a dragon by Merlin by infusing her with the blood of a dragon, and one is shown in Realta Nua as a representation of her Magic Circuits and is visible standing near her Magic Core.
However, it seems that Saber is linked more to lions than dragons:
Though she once felt more linked to dragons, she came to enjoy lions after taking care of a lion cub for a month at some point in her life. She claims that it is not that she "likes" them, but rather the happiness she experienced as he became attached to her brought forth a bond with them. He was very energetic, often biting or scratching, but she wished to be able to stay with him until the end. She has had feelings for them since then, and even remembers the fond memories while holding a stuffed lion. Shirou seeing the memory of her rubbing her cheek against the lion comments on it being a time where she looks like a girl her age.
If I do an image search on Sankaku Channel with the tags
saber lion, I get 129 results, which is much higher than the 31 results that I get if I search with the tags
saber dragon. A glance through the images reveals that, in addition to the stuffed lion in the original Fate/stay night, there are also Saber Lion, the Unit Lion race car that appears in Carnival Phantasm, and a grown lion often found to also accompany Gilgamesh and Iskander.
It does not appear that Saber taking a liking to lions is central to the plot, so why the stronger association of Saber with lions than good old dragons? Was it ever revealed (e.g. in an interview) why Kinoko Nasu chose to portray Saber more as a Lioness than a Dragon Lady?