Great question! Tigers (and dragons, which is important to understand here) have a very special place in Japanese history, intrinsically tied to China, religion, philosophy, and all things esoteric. The exact nature of the historical transmission up through Edo, unfortunately, is not so clear. More generality than anything.
Let's begin in China. The tiger, as you're probably aware, is one of the twelve signs of the Chinese zodiac (jp: tora 虎 ). It represents one of the shisho 四象, the four mythical beasts representing the four cardinal directions, colors, seasons, virtues, emotions, even one of the four elements. Tigers symbolized the west, autumn, and metal element. This becomes important later. For now, understand the importance of the tiger by its relationship to the dragon via the I Ching (circa 400 BC).
云从龙。风从虎 Clouds follow the dragon, winds follow the tiger
(note that 龙 is the simplied form of 龍)
The dragon and tiger are opposites- dragon in the east, tiger in the west. The dragon is typically thought to represent heavenly virtue- which it does- but the idea of "heaven" goes back to the ancient days of the Shang and earlier when it brought the life-giving spring rains for crop growth. Hexagram 1 ䷀ (six solid lines) is a pictorial representation of the dragon's awakening from winter hibernation. Dragons lived in pool and were stirred to action with the first thunders and offerings from peasants back in those days. The literal ascension from the depths of their pools to the heavens for rain is very likely the origin for its association to heaven (ten/tian 天). And as rulers in China sought to use the dragon as symbolism, so to did its "heavenly virtue" grow. Rulers dispensing blessings from above was not unlike the dragon bringing rain from the sky.
The tiger, as far as I can tell, does not have such a cool story. Its known for power and quickness, as is evident by its usage in military aspects of ancient China. For example, Tai Kung's military classic 六韜 (jp: Rikuto, six "bags" though the "bags" here is a play on "secrets" or "hidden talent"...) has the Tiger collection 虎韜, which focuses on military equipment and troop tactics, different from the Dragon collection 龍韜 which focuses on command staff, structure, and strategy. In opposition to the dragon's role in heaven, the tiger finds its home on earth. Clouds in heaven follow the dragon, winds on earth follow the tiger.
Historically speaking, the dragon rituals alluded to above persist with varying levels of belief even through the Tang dynasty in the 8th century CE. Tang poet Bai Juyi wrote a satirical poem on the practice, illustrating the more "enlightened" looking down upon such superstitions. This is important because it ties into what likely are the origins of the tiger in Japan.
Japan as a state doesn't really form until the Yamato era 大和時代 in the 3rd-7th centuries AD. It's at that time when the Japanese "state" becomes more unified as tribal chieftains are brought into the fold. Polities seek acknowledgment of rulership from China via envoys. I alluded to that a bit in this answer to another question on the imperial family origins. As stated there, the Hou Han Shu 後漢書 and archaeological evidence date Yayoi emissaries from the land of Na as early as 57 CE (Japan was called Wa 倭 at the time). Such investiture was important for a would-be ruler in Japan. Yuryaku would be the first "King of Wa" (Wa no o 倭之王 ) if you believe his letters to China. In reality, he was one of five paramounts with whom the Liu Sung dynasty maintained diplomatic contacts with in the 5th century- last of the five to send an emissary, but third to receive recognition.
With recognition was cultural exchange. Buddhism, daoism, Confucianism, yin-yang theory (jp: onmyodo 陰陽道), etc make their way over. It's said the progenitor of the Fujiwara household, Nakatomi Kamatari, had the Rikuto mentioned above as his favorite book. Symbolism for dragons and tigers are included in this transfer. The tiger is found in old tombs as a symbol for the western direction as an example. From a Buddhist perspective, a panel of the 7th Tamamushi shrine depicts the Mahasattva Jataka, a tale wherein the Prince Mahasattva, an earlier incarnation of the Buddha, sacrifices himself so that a starving tigress and her cubs might live.
Tigers continued to appear in the service of Buddhism in the centuries following. From the 14th century, tigers were frequently subjects in Chinese ink paintings imported warriors and Zen Buddhism emissaries. This piece by the Chinese monk Mu Qi speaks to the tiger's power, "when the tiger roars, the winds blast" (虎轟風爆... these kanji may be incorrect as I'm trying to decipher the painting in small pixels, sorry!). That particular piece is referenced as being part of a pair with, you guessed it, a dragon. The idea of the tiger as a symbol of power, quickness, and aspect of change was not lost to Japanese even if they hadn't actually seen one.
It's also been suggested by historians that popularity amongst the warrior elite stems from its daoist origins. Much like the Emperor represented the dragon, raining down blessings from the clouds, the choice of ministers or, say, a shogun, represented the tiger bringing the wind to the earthly realm. Tiger paintings, linked as they were with shogunal collections in centuries past, functioned as talismans of "legitimate" political power in pre-Edo Japan. Military command in the realm of man.
From what I've read courtesy of art professor Richard Wilson, the first live tiger on Japanese soil didn't occur until Toyotomi Hideyoshi's time (16th century) when Korea was invaded. A tiger was captured and brought back to Hideyoshi as an example. But from the early 1600's, tigers were somewhat accessible to the urban population. Live tigers are reported at the riverbed entertainment district in Kyoto in 1648 and again in Kyoto and Osaka in the Enpo era (延宝 1673-1681), although there is some question as to whether these were real tigers or not. A record of the Genroku period (元禄 1688-1704) quotes the sideshow barkers announcing the arrival of tigers from Holland.
Be aware that Japan was effectively "closed off" to the world during the Edo period. From an artist perspective, this meant that tiger paintings were based off historical awareness of previous works in Japan, hence why the style of the tiger is very similar through the middle of Edo in the 18th century. The 1700's did bring some new imports, including books by the Dutch which supposedly offer more detailed depictions of tigers and nature as a whole. However, painters don't seem to have changed their style much. As one painter put it in 1755-
When I paint natural phenomena, depiction is impossible without a true model. As there are no ferocious tigers in this country, I could only indicate the appearance of one by copying [Chinese painter] Mao I's painting
Schools of painting which focus on tigers emerge and continue into late Edo, prior to your Meiji timeframe. Famous painter Kishi Renzan 岸連山 produced a number of pieces, based on those of his teacher and adoptive father Kishi Ku 岸駒 (sometimes called Ganku) who produced tiger pieces of his own. Kishi Ku even received a tiger head from a Chinese resident of Nagasaki in 1799!
Hopefully this helps. Key piece is to understand the symbolism surrounding the animal and the cultural transfer that happened as Japan emerges as a state. Plus, legit live tigers in Edo!
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