What qualifies as a tense or not depends on your definitions of the term. Different linguists and traditions will have different standards and what is taught in school is often not the terminology used in scientific description - it's actually very common for school teachers to teach things that any linguist would think was downright wrong. But terminology is a choice, not something where it really makes sense to say "is" or "is not", the question is how clear does your description end up. (And as always, when you argue about whether or not something is an X, you're not so much talking about the thing as you're talking about the definition of the category X.)
I studied Middle Eastern languages (though mostly Arabic and Persian) and linguistics at a university in northern Europe, and we would treat tense, aspect, and mood as different categories. Often they are distinct and verbs are conjugated both for time and e.g. evidentiality and thus it is fruitful to have two categories. I think this is the case for Turkish, e.g. see how Wikipedia lists the conjugations[0] here as a two-dimensional system. The article uses the term tense (explicitly 'for simplicity'), but I think it makes sense to have different names for the different categories - so tense would refer to the rows in that schema, and mood would refer to the columns.
[0] https://en.wikipedia.org/wiki/Turkish_language#Verb_tenses