I actually think this is a possibility even with the Iberian Union still happening.
The Portuguese discovered Taiwan in 1544, being the first Europeans to visit the island. This was long before the Iberian Union. If the Portuguese had shown interest in colonizing the island, the Spanish would likely have offered no complaints, especially given that Taiwan was on the Portuguese side of the Tordesillas line (which, while far from absolute, was still a powerful negotiational device for the Portuguese). I'm not sure weather on not it would be feasable to have a Portuguese Taiwan before 1580 (the date when Philip II of Spain became Philip I of Portugal, beggining the union of the crowns), however, if colonization happens relatively early during the Unions tenure (somewhen during the reigns of Philip I/II, or maybe even in the early days of Philip II/III) Portugal can still gain undisputed sovereignty over the island, as it still retained quite a lot of autonomy for some time in colonial matters.
Now, unless we butterfly the Dutch away from existance (which I think can be done with a simple PoD set during the partition of the Habsburg Empire, but would cause massive butterflies), they will be a problem, and it's true that the Iberian Union dimished Portugal's capacity to fight back agaisnt their offensive. However, there were plenty of territories that the Portuguese managed to defend even during this period, including Timor and Flores in Southeast Asia and Macau, which quite close to Taiwan. Worst case scenario, the Dutch will have the opportunity to take over Taiwan in the 1620s. By then, the Portuguese will have had plenty of time to set up their presence on the island, and it's feasible that they resist, even with the increasingly neglecful attitude of the Spanish crown towards Portuguese issues.
With the Portuguese in control of Taiwan, they would be able to block the passage of Dutch ships to Japan. This will force the Tokugawa to continue the Nanban trade, even if they restrict it to Nagasaki and still ban christianity and mostly enforce Sakoku. They were pragmatic enough to realize that Japan needed to keep trading at least a little bit with the outside world, and, if they couldn't do it with the Dutch (who they prefered do to their more bussiness-like approach and not trying to spread their religion), they'll just have to stick up with the Portuguese.
It's fascinating to imagine what kind of society could emerge in Portuguese Taiwan, or
Formosa, in this scenario. The Portuguese would send some European colonists to the island (certainly more than the Dutch), but their numbers wouldn't be very significant. They would probably try to supplant that with mixed-race settlers from several parts of Portuguese Asia. These populations were catholic, spoke creole languages, and were very loyal to the crown. Kristang from Malyasia, Topasses from Indonesia and maybe even some Portuguese Burghers from Sri Lanka would likely come to form a new Eurasian elite to rule over Taiwan. Now, as for where they'd get their workers from, that's another story. Importing slaves from Africa is obviously out of the question, and East Asians wouldn't allow for the enslavement of their own peoples, so I trust that Portuguese would realize that filling the colony with Chinese and Japanese slaves would lead to its demise... Well, actually I don't quite trust it, but we need it to happen for the sake of the scenario. Importing slaves from Malaysia and Indonesia would probably be doable, and the Portuguese would try it in the beggining. However, I'm unsure if China and Japan would allow for Portugal to keep a slaves colony that close to their territory. Also, this might lead to the mixed-race elite feeling a bit unconfortable, not being quite sure of what really separates them from those that are enslaved. I think it's most likely that the Portuguese would abandon the slavery option relatively soon and start looking for cheap paid labourers. The easiest place to get those would be China, that's where the Dutch got their cheap labourers IOTL after all... However, Portugal would probably recruit its Chinese labourers from the south of the country, near Macau, where they have a stronghold, so I think it's likely that Portuguese Formosa's Chinese community would be mostly Cantonese speaking.
Aboringinals would be converted to Catholicism and would also oviously form an important part of the island's population. Assuming the Portuguese stay in controlof Nagasaki, there will also likely be a fair amount of Japanese Christian refugees coming to Taiwan, including people from various social classes.
Also, we should naturaly expect a lot of miscenegation over the time.
In the end, I believe that Formosan culture would eventually come to resemble the Macanese one, add only a little more Japanese influence and maybe some elements from arboriginal culture. If they develop their own creole, I'd say it might sound similar to
Papiaçam.