I don't know why we see Goddess Durga sometimes sitting on a lion and sometimes on a tiger? Which animal was actually Durga's mount (bahan)? Tiger or Lion?
I would definitely say that given the scriptural sources, the goddess has only the lion as her mount. But please note that iconographical instructions often give descriptions which are not strictly consistent with the stories. Again, sometimes painters and iconographers take liberties. While the Devimahatmya categorically mentions the lion as the goddess's only vehicle, there are beautiful Pahari paintings where she is fighting Mahishasura while riding on a tiger. Famous Sanskrit scholar Alf Hiltebeitel has also remarked that if one goes by the actual representations of the goddess over the centuries, then the lion and the tiger seem interchangeable as her vehicle. Moreover, scholars have also argued that a half-tiger and half-woman figure inscribed on an Indus seal is actually a prototype of the goddess riding on a tiger, thus hinting that the goddess's association with the tiger predates her association with the lion, and dates back to prehistoric times when Hindu mythology didn't exist in its Classical form.
Durga rides on a lion. The story is told in this chapter of the Devi Mahatmya of the Skanda Purana. Durga was created out of the combined energies of Brahma, Vishnu, Shiva, and other gods. And after she was created, the gods have her a variety of weapons and other ornaments:
The bearer of Pinaka (Siva) drawing forth a trident from his own trident presented it to her; and Visnu bringing forth a discuss out of his own discuss gave her. Varuna gave her a conch, Agni a spear; and Maruta gave a bow as well as two quivers full of arrows. Indra, lord of devas, bringing forth a thunderbolt out of (his own) thunderbolt and a bell from that of his elephant Airavata, gave her. Yama gave a staff from his own staff of Death and Varuna, the lord of waters, a noose ; and Brahma, the lord of beings, gave a string of beads and a water-pot. Surya bestowed his own rays on all the pores of her skin and Kala gave a spotless sword and a shield. The milk-ocean gave a pure necklace, a pair of undecaying garments, a divine crest-jewel, a pair of ear-rings, bracelets, a brilliant half-moon (ornaments), armlets on all arms, a pair of shining anklets, a unique necklace and excellent rings on all the fingers. Visvakarman gave her a very brilliant axe, weapons of various forms and also an impenetrable armour. The ocean gave her a garland of unfading lotuses for her head and another for her breast, besides a very beautiful lotus in her hand. The (mountain) Himavat gave her a lion to ride on and various jewels. The lord of wealth (Kubera) gave her a drinking cup, ever full of wine. Sesa, the lord of all serpents, who supports this earth, gave her a serpent-necklace bedecked with best jewels. Honoured likewise by other devas also with ornaments and weapons, she gave out a loud roar with a defying laugh again and again. By her unending, exceedingly great, terrible roar the entire sky was filled, and there was great reverberation. All worlds shook, the seas trembled.
So Himavat, the god of the Himalayas and father of Shiva's wife Parvati, is the one who gave Durga her lion.