How shall the word "biology" be interpreted, if no English word can start with two stressed syllables?

Solution 1:

"No English word can start with two stressed syllables" is just false, unless you define "stressed syllable" as "primary-stressed syllable", in which case it is trivially true (because by definition a word only contains one primary-stressed syllable).

A better rule is that no English word can start with two fully unstressed syllables (two syllables with a reduced vowel).

As you have found, it's definitely the case that an English word can start with two syllables with unreduced vowels. Some theories of English stress allow unstressed syllables to contain unreduced vowels (at least in some circumstances), while other theories of English stress treat any syllable with an unreduced vowel as having at least some degree of stress (theories may make use of a concept of secondary or even tertiary levels of stress).

Solution 2:

You're conflating two things. Stress and vowel reduction. I too pronounce the word as /baɪˈɒl.ə.dʒi/; the only stressed syllable being the second one. The only reduced vowel is the third one. The first and fourth syllables are neither stressed nor reduced.

I believe what your teacher meant by "you understand a syllable is stressed when it's not reduced to a schwa and when it is therefore a full vowel" is that a stressed syllable is never reduced, but this does not imply the inverse, that a non-stressed syllable is always reduced. Indeed just the words "always" and (albeit not always!) "reduced" disprove this!