Before anything, I want to make something very clear.
I never push my birds to do things they don't want to do. My questions about if I'm doing something wrong training my birds Isn't because I'm impatient, even if it takes months or even years for my birds to really trust me, I'm okay with that. I just want to know if I'm doing something wrong.

So, here’s where I am right now with my birds.
My brother had been watching lots of videos of budgies doing funny tricks for a few weeks and after going out for lunch with a friend, impulse bought a budgie (Rivin), cage, toys, treats, and food at PetSmart without our parent’s permission. They had bonded very well, but around 2 weeks after, he moved the cage downstairs because Rivin would wake him up very early in the morning. He began to pay attention to her less and less, and eventually bought another budgie (Kiwi) to keep her company. Now that Rivin had a friend and my brother was paying attention to them less and less, all previous bonds with them were broken and he was less motivated to do things with them than ever before. It devolved to just talking to the birds for about 45 seconds after coming downstairs once a day. A month later, my brother moved out to college. After 2-3 weeks of the birds being in the living room doing nothing but having their food and water refilled every few days, I felt bad because the birds seemed bored and lonely, so I decided to take care of them and try to tame them, as they were very afraid of us. I talked to them softly, tried to feed them treats, and eventually Rivin got used to stepping up onto my finger. After this, virtual school had started and I moved the cage into the room that I spend pretty much the entire day in, it’s where I do school work and stuff, so I’m usually alone with them 24/7. I spend maybe 2 hours each day trying to train them. Sticking my finger out while holding a treat, waiting for them to hop on it, trying to get them on my finger or seeing if they want to be on my shoulder, but the thing is, looking back at 2-3 months ago, I really don’t see much of a difference in the progress I’ve made with them. The difference between now and 3 months ago is the birds don’t panic when I’m near the cage, Rivin is used to being on my finger, Kiwi very rarely steps onto my finger usually only if I’m holding a treat, and they both will eat millet out of my hand. That’s it. After 3 months, that’s all the progress I’ve made. Rivin still bites my hand if it’s too close to her, Kiwi is still terrified of my hand, it seems like the only progress I’ve made with them is that they hop onto my finger because I may give them a treat, or they hop onto my finger because it’s what they think I want them to do, not what they want to do. After all this it feels like they still hate me, and I don’t know what I’m doing wrong. I’m not frustrated with them, they just always seem so scared of me and I don’t know what I can do to show them I’m not trying to hurt them.

I never get my hand too close to them if they seem scared,
I never force them to do things,
I never introduce too many things at once,
I never try to punish them for biting me,
I'm always very patient with them.
But it feels like I've made almost no progress with them. Am I doing something wrong?
Please help me, if you need more specific information about them I would be happy to provide. Thank you.

It doesn't sound as though you are doing anything wrong.
Some budgies just aren't interested in really bonding with people and some are.
The fact they aren't as afraid of you now is great.
I have several budgies and some of them won't let my hand get near them while others will come and sit on me if I'm in the room -- I simply accept all of them as they are.
Do you let your budgies out for free-flight in a bird-safe room?

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I thought it may just be that the birds are just simply the type of birds who generally don't like human interaction as much, however the birds seem quite happy and dislike leaving my finger on some occasions, and take every opportunity to leave my finger on other occasions which confuses me, which is why I asked this question. My birds give me a lot of mixed signals like that. Additionally, Rivin was very social when she was with my brother and enjoyed being on his shoulder, finger, and head. She is much different now.

Yes, while I am home alone (which is the majority of the day) I lock all doors and windows and open the cage in the room I am in. I lock the doors because occasionally the birds fly out of the room by accident and they grab onto a window and sit there not knowing what to do, so I put my finger under them gently until they step up, and I bring them back to the room. They are frequently out of the cage, and enjoy exploring around the room.

It's like Deb says, some birds just prefer the company of other birds to human or it takes a very very long time. My boy, who had been neglected in his first home and was an adult when I got him bonded to me right away and is the friendliest little dude ever. My girl, who I got from a shop as a baby, took well over a year before she would willingly come to me and hang out on me on her own, and even then for her entire life our relationship was 00% on her terms.
Just keep being respectful and loving to them and who knows? One day one or both might decide to hang out with you specifically.

Thank you, I hear a lot of stories about some budgies needing lots of extra time to be comfortable being around people, but since I had also heard a lot of stories saying the opposite, I was a bit confused. Thank you for the advice, I really appreciate it.

Although it isn't the same type bird, one of my rescue lovebirds (Pedro) is very finicky about when and how he wants to interact with me.

I've had Pedro now for over 9 years and only in the last year has he felt comfortable enough to sit on my shoulder, jump on a food dish when I hold it for him, etc.
He *still* refuses to be touched and will try to bite if one puts their hand near him or tries to get him to step up.
I just give him the freedom to determine how and when he wants to interact with me and I've been thrilled that after all this time he's finally become more trusting!

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Just to echo what everyone has already said, we have 3 parakeets who are very happy to be on their own. They will feed off my hands—what my wife calls "cupboard love". They leave the moment the food is gone. I'm just happy they don't seem to mind us sharing the room with them. :-)

I think that 2 budgies is much harder to deal with than one, and more so if you left them there for a long time without any interactions.
(Personally, I've seen that some budgies tend to stay loyal to one person. A family friend gave me a 7 year old budgie a few years back, the budgie has now passed away, and he refused to do anything with me, including stepping up)
Good Luck with taming them!
Also they make a good alarm! (I tend to hit snooze a lot and their chattering wakes me up a lot better than any alarm)

It’s quite possible that being abandoned by your brother after bonding with him has made poor Rivin distrustful and hurt. Birds are emotional and very intelligent creatures, and having him essentially make her fall in love with him and then leave her could have hurt her deeply. Give them all your love and don’t expect anything in return. They may surprise you. ❤️