Simulating the sound of analog tape
Interesting story, which I once read in a book about Pink Floyd: When Roger Waters was recording Welcome to the Machine in 1975, he lamented that he could not capture the sound of the synthesizers he was recording on the then state-of-the-art analog tape decks. He was able to minimize how much the recording equipment affected the sound of the synths by bypassing the mixer and directly connecting the synths to the tape deck, but it still affected the sound, making it sound like “it was going through a PA system” or words to that effect.
I am of the “digital recorded at 44.1 is indistinguishable from a wire” crowd . This means that, given a decent converter (the cheapest computer interface which feel has a decent converter is a Focusrite 2i2, but I haven’t had a chance to try the PreSonus AudioBox or the Behringer U-Phoria), and a reasonably set level, the conversion to digital will put no coloration whatsoever on the signal. What goes in when I monitor the signal is identical to what goes out after the signal is recorded.
That said, yes, I find the sound of digital a little harsh in the upper midrange. I’m used to the tape effect where upper frequencies are attenuated.  To simulate this with digital, after mixing down a song, I like to run it through an equalizer which gently rolls of frequencies starting at around 1khz. This takes some of the edge off without sacrificing the clarity of digital.
If trying EQ and it doesn’t work: Maybe try one of the “virtual tape recorder” plugins. I have heard good things about Slate VTM, but it’s a pricey $150 plugin and uses the obnoxious iLok copy protection which many do not care for. But, maybe that will give one the sound the warmth missing when recording digitally. Another good plugin is U-He Satin, which has less obnoxious copy protection.
Another option, of course, is to get a used analog tape deck, either a 1" or even, if given the room, 2" 24-track, but that is a good deal more expensive than most interfaces, a maintenance pain in the butt, and the tape is expensive, especially if it’s 2" tape. 
 I prefer recording at 24-bit, simply because I have to worry about setting levels less when I have the 12 db or so increase in signal to noise ratio 24-bit gives (in theory, 24 bits gives me 48db less noise than 16 bit. In practice, no one makes a converter that quiet; not even the PCM4222, which is probably the best converter money can buy).
In more detail: I am used to, when hearing a recording, having a high frequency roll off in the recorded sound. When that roll off is gone, the recorded sound sounds unnatural and harsh to my ears even though, yes, it’s a more accurate representation of what is being recorded. I find removing some of the high frequencies restores how I expect a recorded song to sound like. Especially since the electronic music I like (Vangelis, 1980s Jonn Serrie, Brian Eno, etc.) was recorded with the limitations of analog tape.
 It looks like, glancing at reverb.com and ebay, that 2" 24-tracks are more readily available than the more compact 1" 24-tracks which were briefly in vogue in the early 1990s before the ADAT killed the project studio analog tape deck.