Actually, I want to add another layer of reasoning about why Alastor doesn't like labels and doesn't directly call anyone a "friend." And to confirm once again that perhaps Alastor is not as averse or incapable of friendship and close relationships as is commonly believed by the fandom.
I'll start from afar - in fact, we know that Alastor has always preferred female company, he likes to communicate and spend time with girls in a friendly platonic sense.
BUT
He also lived in a time when friendship between a man and a woman was something unthinkable. Even in our time, there are still ongoing discussions about whether friendship exists between a man and a woman.
And it's likely that almost any girl he wanted to talk to/spend time with automatically assumed either a romantic or sexual context. And moreover, the people around them, too, seeing Alastor's friendship with the same Mimzy, probably thought that they were together in a romantic or sexual context.
All of this is united by the fact that other people thought they knew how him to live and how to feel better than him.
In fact, probably of him was constantly given a label that contradicted reality and put pressure on him to adjust to their perception. He can constantly encountered misunderstandings of his intentions and expectations from society or other people. At the same time, with the feeling that this perception of ALASTOR is not correct from a public point of view.
I'm not saying that this is 100% why Alastor avoids labels. It may well be related to completely different things, but I also like this interpretation. This may be one of the reasons.
If he said "friendship" and the "relationship/sex" paradigm was imposed on him and it was impossible to be friends with girls, he could well end up just freaking out and refusing to use any labels in any cases. Just to refuse to give other people the opportunity to define his reality instead.
Thus regaining the freedom to interpret your relationship with someone. In the end, this could well become not just a trauma of attachment to an example, but also a fundamental issue of freedom, control and autonomy.
Because it seems that he is quite happy to communicate with someone in an obviously friendly way for years without any concern - he communicated with Vox for several years quite calmly, relaxed and happily until he offered a partnership, and in that situation, for the most part, Alastor was hurt by the fact that he thought Vox wanted to use him.
Besides, there was nothing in Vox's entire speech that bothered Alastor before the offer of partnership and the outstretched hand, the entire opening speech of Vox BEFORE that was normal for him. He agreed with Vox's other statements.
"Well, look, I'll just get to the point. We've been close for a few years now, right? I mean, people know us, the love us. And with new overlords popping up every day. And before you hit me with a "Well, you're pretty new yourself," I know, okay? But I'm much more forward-thinking, so it's in your best interest to hear me out."―Vox-box, to Alastor
And at this point, Alastor did not object in any way
He's been talking to Mimzy for decades. With Niffty, as far as I understand (?) although there are no exact dates, it is clearly years. Alastor is communication with Zestial, although it's not a fact that they communicate very often or closely, but Alastor is friendly. It's debatable with Rosie, but the fact having her in Alastor life should be mentioned.
In general, it seems that Alastor is not embarrassed and he is quite good at communication, even quite close. And it is the intersection of some borders that triggers him. He is an avoidant yes, a very avoidant person who hates showing vulnerability, hates losing control, fiercely guards his freedom and autonomy as one of his main principles, yes he does not like labels.
But maybe he's not as avoidant as it might seem at first glance :3
