I second both Tim and El Ectric: There's no harm in using just intervals for this purpose, but I also doubt that it's your main difficulty. There are many questions about ear training on the site with lots of good practical information, but there's one bit I don't see represented in them. I would question whether the electronic audio tools you're using are the most effective for you.
I find that timbre matters a lot to me for pitch identification. I have "acquired pitch"; from playing violin for years, I can recognize pitches that I hear, but have more accuracy in the violin range, and have difficulty with some instruments like saxophone or the human voice. I also have difficulty identifying pitch from pure sine waves—which is what a lot of apps use.
The quickest thing I'd suggest is trying other apps that might use different sample banks and seeing if it's easier. But there could be some good reasons to move to an actual analog instrument. Real instruments, like a piano, have complex waveforms rich in overtones. An excellent sample might do a great job of capturing them, and excellent speakers might do a great job of reproducing them, but something will always get lost in translation. Using an actual acoustic instrument will give your ears and brain more data.
Further, you'll be "playing" an additional instrument: the room. If you've been primarily training with headphones, then you're vibrating very little air. An acoustic instrument interacts with its space, and the acoustics of the room become additional data for your senses. This is especially important when training intervals. When two pitches are played simultaneously, their sound waves collide and interact, canceling or reinforcing each other. They can combine to create new pitches. If you, say, simply play one pitch into one side of your headphones and one into the other, then these interferences are never actually happening in the air, just in your perception. Even if you blend the two pitches in headphones, that's still a much quieter sound in a much smaller space; a low-energy wave in your ear canal rather than more energy in the air of the room, with more chance for the waves to interact.
Finally, identifying a simultaneous interval is much harder than identifying a sequential interval (i.e. play the first note, then the second note). Make sure you've mastered that before moving on.
Of course, there are some impracticalities: to play notes on a real piano and guess what they are, it's best to have another person play them. Even if you don't look at the keyboard, you might have a sense of how far apart the keys are. So if you don't have a piano and assistant handy, you could get creative. You could get hold of a pitch pipe like this one:
... and, without looking at the top, play a note, give it a spin, and play another note. Or even simply unplugging the headphones and using good speakers for apps or websites might be a step in the right direction.
It can also be very helpful to simply play notes even if you do know what you're doing. Sit down at a piano and really intentionally study the sound of minor thirds; play them in various ranges, over and over, and listen intently. Pursue playing an instrument. Ear training is usually done in support of making music, and every time you play a note you "smuggle" an awareness of it, and of its relationship to its neighbors, into your subconscious.
