Calling out the notes is not required on Rosh Hashanah as far as I know.

Why do you limit yourself to "there would seem to be no need to blow the Shofar during Elul without a Minyan"? All your arguments apply equally to a minyan as well

Calling out the notes is not required on Rosh Hashanah as far as I know. - Strictly required, no. If someone is blowing Shofar for an individual or group (or obviously, for himself by himself) outside of the regular Davening, calling out the notes is not needed. But it is a very structured and (I believe) important part when Shofar is blown as part of Davening. But that is not done during Elul - i.e., a definite difference from RH.

The difference (but I don't have the solid proof - if someone can provide it that would be great, I don't have time to research it right now) is that on RH it is normally with a Minyan but the obligation - particularly as evidenced by the Brachah - applies even without a Minyan. On the other hand, during Elul I believe this is a "part of Davening with a Minyan", not to require a Minyan but rather something that goes "with Minyan Davening* and not a personal obligation.

Calling out the notes is practiced on Yom Kippur at Neilah in at least some communities, even though that blowing is 'only' customary.

This answer feels to me to be entirely missing the point. While clarifying what level of obligation hearing shofar during Elul is (and a custom can be obligatory, despite the apparent word games in this answer), the OP still doesn't know if a woman fulfills it if she blew the shofar herself. (Not to mention that the most important argument here, that of the apparent significance of a minyan, is entirely hidden and unsupported.)

@AA End result: 1 - I think but don't have supporting sources and don't have time to do the proper research right now that this is truly not obligatory and therefore not a question of "if a woman fulfills" but rather nothing to fulfill; 2 - Feel free to write a better answer - if sufficiently better, I will delete my own.

If one doesn't have a correct good answer to post one can't feel something is wrong with what is posted? I very much dislike deflecting legitimate criticism with "I don't see you doing it better". That may not have been your intention but that's how your suggestion comes across to me.

@AA That wasn't my intention. But I still believe my answer is correct, until shown otherwise. Good is in the eye of the beholder. I had a somewhat opposite situation on Electrical Engineering - I saw a serious flaw (obviously, that is subjective - you see flaws in my answer here that I think are not a big deal!) and after a few rounds of comments I went ahead and wrote my own answer.

@AA I am also perfectly fine (I have discovered both here and on SE/SO that some people are OK with this, some are absolutely not ) with you or anyone else adding/editing useful information to my answers.

Sources noting that the custom does not apply without a minyan https://hebrewbooks.org/pdfpager.aspx?req=1081&st=&pgnum=304 https://hebrewbooks.org/pdfpager.aspx?req=14511&st=&pgnum=150

@manassehkatz Sure. I've made a suggested edit

The question is asked why isnt it bal tosif.