From a certain point of view, both the first and last syllable of "Portuguese" are stressed. The variation lies in the placement of the "accent" or primary stress (which is associated with a higher pitch in English).
Even for a speaker who puts the primary stress on the last syllable of Portuguese, there will be some stress on the first syllable (a “secondary stress”). And a speaker who puts the primary stress on the first syllable might be said to have some kind of stress on the final syllable, because it has an unreduced vowel. (In certain theories, this would be called "tertiary stress".)
There are a number of words in English that have shifted an originally secondary stress to a primary stress, or vice versa.
I don’t know of any reason to categorize the pronunciation with primary stress on the final syllable as “wrong”.
The Oxford English Dictionary (OED) suggests that putting the primary stress on the first syllable of Portuguese is characteristic of an American English accent:
Brit. /ˌpɔːtʃᵿˈɡiːz/, /ˌpɔːtjᵿˈɡiːz/, U.S. /ˈpɔrtʃəˌɡiz/
But I am an American English speaker and I pronounce it with the primary stress on the last syllable.
Other 3-syllable words that can be accented on either the 1st or last syllable
Another word that seems to show similar variation is manganese, which the OED says may be accented on the final syllable in British English.
Trisyllabic words not ending in -ese that show variation include parmesan, millionaire, cigarette, obsolete, and various words ending in -ine /iːn/ such as magazine, gasoline, submarine, limousine.
parmezan: OED says final accent in British English, final or initial accent in U.S. English.
millionaire: OED says final accent in British English, final or initial accent in U.S. English.
cigarette: AHD shows both stress patterns for U.S.
obsolete: AHD shows both stress patterns for U.S.
-ine /iːn/ trisyllables. Magazine, gasoline, submarine: OED shows both stress patterns for both U.S. and Britain. Limousine: AHD shows both stress patterns for U.S.
In certain contexts, any -ese word may be pronounced without an accent on the final syllable
There is one regular phenomenon that may contribute to such variation between pronunciations with the accent on the final syllable and pronunciations with the accent on the first syllable. When a word that has stress on the final syllable comes immediately before a word in the same phrase that has an accent on the first syllable, the first word may experience "stress retraction": the stress on its final syllable is deleted, causing an earlier stressed syllable to be realized with the "primary stress" of the word. This means that even speakers who say he is ˌPortuˈguese might say the ˈPortuguese ˈGovernment. (Likewise, "the Japanese government" might be pronounced without an accent on the final syllable of "Japanese", and same for "the Chinese government"/"the Chinese language").
I don't know of any other -ese nationality word that is like this
Although there are other examples of three-syllable words with similarly variable stress patterns, I don't know of any other -ese nationality word in particular that shows the variation that evidently exists for Portuguese.
So far, I have not found any dictionaries that provide evidence of speakers who always put the accent on the first syllable of Japanese, Cantonese, Javanese, Siamese, Lebanese, or Taiwanese.
I can only speculate as to why Portuguese seems to behave differently from the other words; if I had to guess, I would say that it's possibly relevant that it's more frequent than most other three-syllable -ese words, except for Japanese; and speakers might avoid placing the accent on the first syllable of Japanese because the monosyllable "Jap" has historically been used as a slur.