I must answer the question but I am not sure how to understand it... The question is: why nasals both can and cannot be treated as stop consonants? I thought that nasals cannot be stop consonants almost by definition, now I am so confused!
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It's all about the manner of articulation.
A stop consonant is by definition a sound produced by the complete obstruction of airflow though the mouth, at least for a short time. There are two kinds, oral stops (what you call "stops"), which keep the velum raised, preventing airflow from escaping though the nose, and nasal stops (what you call "nasals"), which lower the velum, allowing air to escape though the nose. Because nasal stops completely obstruct airflow though the mouth, they are, in fact, by definition stop consonants.
However, because there is air flowing out the nose, a nasal stop can be "continued", unlike an oral stop, which can only be heard and distinguished upon release. Because a nasal stop can be vocalized indefinitely, it can be argued that nasal stops aren't really stops. This is probably why they are often simply called "nasals".
The answer can be either yes or no—it comes down to your definition.
Some people define "stop consonants" to be consonants where the airflow is completely stopped (as in, the opposite of continuant consonants). In this case, nasals are not stops: the airflow continues through the nose, which is why you can extend them and keep the sound going.
Other people define "stop consonants" to be consonants where the airflow through the mouth is completely stopped (as in, the opposite of fricative and approximant consonants). In this case, nasals are stops: there's no airflow through the mouth, only through the nose.
Neither of these is more correct than the other: it just depends which definition you (and your instructor and your textbook!) are using. The second definition is more popular, in my experience, but e.g. Ladefoged uses the first, and it's hard to argue that The Sounds of the World's Languages isn't a solid precedent.
This being an obligatory answer to a question, we would have to know your instructor's ideology and instructional point – i.e. in the present instance, we can only offer reasonable interpretations of what he/she might have had in mind, based on what is actually said and done in linguistics.
The phonetic term "plosive" unambiguously refers to [p,t,k] and not [m,n,ŋ]. "Stop" is a more variable term, which can refer to "occlusive" (including nasals) or "plosive". In addition, in the context of a theory of phonology with distinctive features, "stop", "fricative" and so on are informal terms describing certain feature specifications. [p t k], which are called stops, would be [-continuant], and [sʃx], called fricatives, are [+continuant]. In that context, the question would be whether [m n ŋ] are [+continuant] or [-continuant].
The feature continuant is defined in three competing ways. The classical definition of [continuant] in Sound pattern of English is
In the production of continuant sounds, the primary constriction in the vowel tract is not narrowed to the point where the air flow past the constriction is blocked; in stops the air flow through the mouth is effectively blocked.
By that definition, because flow of air through the mouth is blocked (occluded), nasals would be [-continuant], therefore informally called "stops". The feature could be redefined in terms of what is blocked, so if [continuant] is defined as continuous air flow through the supraglottal cavities, nasals are [+continuant] but [ptk] are still [-continuant]. You can also define [continuant] in terms of airflow through the midsagittal region (central channel of the oral cavity), which renders [ptkmnŋl] "stops" (i.e. [-contniuant].
As for the "can and cannot" problem, that could refer to competing theories of the definition of the feature [continuant] – if you adopt the supraglottal air flow definition, nasals are not stops, otherwise they are. It has been held in the past that there is a fixed definition for these features, so that we might have different ideas about how to treat nasals and plosives, but there aren't actually two ways to treat nasals. But a different metatheory of features is that they do not (contrary to the SPE tradition) have narrow phonetic definition, and the features of [ptk] vs [mnŋ] might be language specific. For example, there might be a general definition frame for [continuant], but also a variable term which has to be set for a given language. In that case, it would simply mean "in this language, [m] is treated as a stop".