Tense vs. aspect vs. mood
Let's first clarify what the different categories mean in the first place:
- Tense is a category that locates events on a timeline. Distinctions between different tenses are often described by means of relations between event time (E), speech point (S) and reference point (R).
For example, when I utter something like I realized that I had forgotten my wallet and we consider the embedded sentence I had forgotten my wallet, the speech point is now, the reference point is the moment in the past where I had the realization, and the event time is even further in the past, namely the point where I lost my wallet.
- Aspect is a category that specifies the view taken on an event: An event can either be viewed from an outside perspective, perceiving the event as a completed whole, or from an inside perspective, perceiving the event from within as something continous.
- Mood is something rather different: Roughly, it signals whether, e.g., the event described is a fact, a command, a possibility, a counterfactual proposition, a reported speech, etc.
Perfect vs. perfective
The difference between perfect and non-perfect can be illustrated by a pair of sentences like
(1)-a The military have taken over leadership (perfect)
(1)-b The military took over leadership (non-perfect)
While the perfect sentence indicates the action is still having an effect on the reference point (the miltary is still in power), this does not follow from the non-perfect sentence (the military took over leadership at some point in the time, but the resulting state might not extend up to today).
Velupillai (2012) treats perfect as an instance tense and not of aspect, under the assumption that it is the defining property of the perfect to express than an event is still relevant/having an effect to some later location on a timeline, thereby making the event span between two locations on a timeline:
Under this account, perfect is be considered a tense, rather than an aspect, because it has the function of locating an event on a timeline - more precisely, locating the event to be spanning the time between the points E and S/R.
(Here, S and R coincide - however, this holds only true for the present perfect where the speech time is the same as the reference time (as in They have taken over); in past perfect (as in They had taken over), the reference time is before the speech time, and the event would extend only until R, not up to S).
In contrast, perfective is considered an instance of aspect, indicating that the event is being viewed as a "bounded whole" (from an outside perspective), rather than as an ongoing event from an inside perfective (which would be called imperfective):
The difference would be between sentences like
(2)-a I cooked lunch (perfective)
(2)-b I was cooking lunch (imperfective)
In the perfective sentence, the event is reported from an outside perspective as a bounded event, while in the imperfective sentence, the event is being viewed from an inside perspective as something ongoing.
Velupillai makes a sharp distinction between perfect and perfective here, arguing that the former is an instance of tense (because it locates an event on a timeline) while the latter is an instance of aspect (because it describes the view taken on the event). However, she also notes that perfect is often being labelled as an aspect category.
The problem is that English (or French) does not actually express aspect independently: The distinction between perfect and non-perfect, and perfecitve vs. non-perfective becomes apparent only in combination with tenses, such as past perfect, past perfect progressive, simple past, present progressive, etc. This makes it hard to actually draw the line between what is an aspect and what a tense, especially given that in school, different "tenses" (which might now be arguably pure tenses or mixtures of tense and aspect) are beingt taught in the shape of the complex conglomerates mentioned above.
Note that, for example, Wikipedia introduces the term imperfect as "a verb form [...] which combines past tense and imperfective aspect". I think it's pretty much the same issue with the name "past perfect": It combines a notion of tense with a notion of aspect.
Especially in as far as the difference between perfective (what we considered a tense above) and imperfective (what we considered an aspect above) in English is concerned, the difference is not immediately obvious: In a situation like
(3)-a John has lived in Chicago for five years
(3)-b John lived in Chicago for five years
I would argue that it is both tense and aspect that make the difference in meaning:
In (3)-a, the use of "past perfect" indicates that John's living in Chicago is a still ongoing state of affairs, there while in (3)-b, the use of "simple past" (non-perfect) suggests that the situation described does no longer hold true in the present, which is a tensual concept (supporting the view that perfect is a tense category).
However, at the same time, viewing (3)-b as something that describes a terminated event comes with a strongly perfective notion, as opposed to (3)-a, where the event is not really regarded as a bounded whole, which is an aspectual category. (Note how things are twisted: The sentence that doesn't use past prefect tenes is the more perefective one!)
This would lead to the conclusion that the use of perfect does have an aspectual component, and that it is not simply a matter of tense.
And now what?
What this all boils down to is that it might not be wrong to categorize perfect as an instance of tense, by assuming a difference between perfective and imperfective where the latter as a form of aspect and perfect is a device that locates events on a timeline (in the sense that some event is still of relevance to the speech or reference time), in the style that Velupillai treats these two terms.
However, what the above discussion also shows is that this distinction is not as straightforward when interpreting how different "tenses" are being used in English (or French or whatever other language), and that complex combinations of tense, aspect and possibly mood are being subsumed under the term "tense" because it reflects how aspect is being used in that language - namely in combination with tense categories, and not as an isolated grammatical function.
And because it is simply easier to explain to students. Telling them they are going to learn a new tense is easier to grasp than starting to introduce terminology that even linguists disagree about. Up to now, I haven't seen a word of command that puts an end to the discussion of perfect vs. non-perfect, perfect vs. perfective, tense vs. aspect once and for all, because such complex abstractions are vague by nature and languages don't always behave exactly in the categories you want them to.
"Perfect" can be considered a tense if one distinguishes it from "perfective" and considers perfect as soemthing that locates events on a timeline.
Perfect is in reality often, if not usually considered an aspect.
The reason is that the way perfect is being used in English, French etc. is a complex combination of tensual and aspectual concepts, putting into play the notion of perfective vs. imperfective also.
Calling it all tense is simply easier if it's only for the purpose of teaching students and not in the context of detailled linguistic analysis.
Velupillai, Viveka. 2012. An introduction to linguistic typology. John Benjamins Publishing.