Like you and one of the other answerers (as of writing this), in a class I TA for, we use Zoom for lectures and Discord for something else--in our case, we use Discord for office hours (undergrad, grad, and professor). Some of our TAs were the ones who thought of using a Discord server when we first moved everything online, so they ran the idea by the professor and set up the server. It's been 1.5 months now and it seems to work well enough.
Since you asked for thoughts on this, I'll share my thoughts. So this is a little bit rambly. My apologies.
Our particular setup involves having a general voice channel that any student can enter and then multiple voice channels that only the TAs and professor can enter and exit freely. Students must be pulled in by a TA or professor, and we each help one student at a time. The general voice channel + the general text channel, in which students should post what they need help with, are basically our queue. Once we're helping a student, we can stay in Discord, or we can move to Zoom.
Students must also request a Student role before they get access to anything more than the text channel in which they request the role, but we don't have any kind of verification system in place. Theoretically, though, we could have a verification system by having students message us a school ID that all students have that is not particularly sensitive information but is still relatively sensitive: You can easily find this ID for any currently enrolled student at our university if and only if you are also a currently enrolled student or faculty at our university, and the professors and TAs for any given class can see this ID for all students enrolled in that class (but students cannot see each other's IDs in this manner, nor can they see the roster unless the professor makes it visible).
We have posted the link to our Discord server in places where only those who are currently enrolled can directly see the link. So even without a verification system, the risk of having outsiders enter is low, and any outsiders who enter are likely to be students' friends. There's a lot of trust going on here.
We also have a TA Lounge voice channel for just the TAs. Since we have multiple TAs during the same OH, I find that this channel helps to preserve the sense of camaraderie that we had before moving everything online, especially on slow days :) I love TA bonding time.
Even though some of us TAs are technically online all the time on Discord, we only help students with stuff like homework during our office hours, and we make sure students know that. We answer logistical questions whenever we see them, though; I think we've been treating those questions as if they were posted on Piazza, which we check at various times.
Some students are clearly using their existing Discord accounts, and a small number don't even have any identifying information in their display name or Discord tag/ID. I personally don't see any issues with students using their personal accounts or even not displaying personal information as long as there is nothing obviously problematic like an offensive avatar or name. So far, we haven't had any issues like trolls or unprofessional behavior, and if there are any outsiders then they've been keeping quiet. Due to our lack of verification, there is the risk that a casual, low-tech bad actor can come in and get a bunch of the students' real names or private message other students and pretend to be nicer than they actually are or do some other social engineering, but I think this is basically the same risk as having a Facebook or something. (I say that as someone who refuses to have a Facebook account.)
It's a little different among the TAs who have existing Discord accounts, although all of us have our real names as display names. Some TAs are using their existing accounts. I personally made a new account because of the reason you listed in the question; I think "mixing business with pleasure" is more relevant for people in instructor positions than for students. On my existing Discord account, I'm in many servers, ranging from game communities to communities of students at my university, and I already feel strange about using the same account for servers where I don't know everyone in real life by definition and servers where I can easily meet people in real life by definition. (I've always wondered about the reaction people will have if we happen to be in both a game server and a "real life" server. I don't have to worry too much since I'm always nice :) But the prospect still makes me a little nervous.)
Something to note: You can't share your screen or see shared screens when you are in Discord on a web browser. I and some students found out that out firsthand in 1-on-1 sessions as well as the one time we decided to have a lecture-style session during OH. This is particularly relevant for computer science classes in general, compared to classes in other disciplines. (I am logged in to my "professional" Discord on my laptop's browser and my iPad so that I can remain logged in to my personal Discord on the Discord apps on my laptop and phone.)
While requiring students to download some kind of app for communication that isn't required by the school (Slack comes to mind) is not unheard of in classes, it feels strange to require students to download Discord, which isn't primarily marketed for professional/business use and which people may already know as a platform for gaming communities and other informal communities. I think what's going on is context collapse. But I suppose there isn't anything that's actually wrong with that. As the cherry on top of the strange feeling, though, it feels weird to require students to download the Discord app when the web browser version exists and is very functional, even though it's missing a feature that is important to students in our class. I'm not quite sure how to articulate why.