I know this post is a couple weeks old, but I thought I'd chime in, as this kind of sound is my jam and I'm always trying out new ways to get there :)

As other people said, a compressor alone probably won't do it.

What I hear in your examples is pretty substantial saturation. My preference these days for this kind of thing is U-He Satin, which is an absolutely amazing tape emulation with just incredible saturation. I can feed a pretty mundane drum loop into it, overcook the input gain savagely, and the thing just instantly oozes vintage breaks and trip-hop vibe. If you don't want to invest in Satin (although it is really a magical plugin), the "A" mode on Soundtoys Decapitator or Soundtoys Devil-Loc (both on sale until Feb 29!) and Klanghelm's IVGI (freebie!) can also definitely do the trick.

One of the key things that really makes a difference when doing this is where in the chain you put the saturation plugin, and what drums are going through it. It's not necessarily obvious at first. In the Tame Impala song, I hear a moderate amount of tape saturation on all the drums. That's simple enough - the drums are probably saturated all together. You can hear the kick pretty cleanly on it's own a few seconds into the song, and there is a very distinctive tapey "crunch" on the attack portion of the kick, before the volume drops for the subby tail portion, which thus sound less distorted. The snare, likewise, is nice and saturated too: instead of a super punchy, transparently recorded acoustic snare drum, we have a fully sausaged sound. In this sort of sound, the punch of the attack comes not from a transient spike, but from the sound of that transient spike being chopped off by tape/desk/converters/plugins.

This is why saturation is key: very few compressors will work fast enough to slam that transient spike. [Here's an example of an analog piece of gear that can work this fast], and has excellent saturation as well. You should be able to immitate most of the dirt, grimy, heavily compressed sounds in this video with a compressors, a good saturation plugin, and a brickwall limiter or clipper.

On the other hand, a lot of trip hop artists will smash their kick and snare, but have a separate layer of hi-hats and/or tambourines that are not saturated like this with the rest of the drums. This maintains some transient life as well as sparkle in the high end.

I like to combine the best of both worlds. For my drums, I'll start by chopping up a breakbeat or two until I get the pattern I want; then, I'll cut off the low end and layer a nice meaty kick and a nice meaty snare. That way, the subs and snare sound modern but the groove still sounds vintage. But I really like the crunchy sound of the hats being chopped along with the kick, so I'll layer a nice vintage hat on top of the kicks and/or snare before sending all of the above into Satin or Decapitator (and sometimes both) with ridiculous amounts of gain. The transients get chopped off, and the kick sounds absolutely massive - but the crunchy part of the kick is really just the layered hi-hat, so the low-end escapes relatively clean. Then, I'll add a second layer of hats in the 5kHz+ region and spread across the stereo field.

The advantge to any of these methods is that you might not even need a compressor. The track I'm working on right now has a boatload of saturation of the drums, but no compression yet - no need for it. The dynamic range is already squashed, and it sounds great (well, it does if you like dirty drums!). The best part is that if I want some more transient punch, I can use a transient shaper or even just a compressor after all the saturation so that the first ~10ms of each drum hit is emphasized.

Another cool thing is that your big cymbals (crashes) won't be pumping due to overcompression, which is normally extremely annoying on the drum bus. Instead of ducking obviously every time a the kick or snare hits, they'll just ring out, warm and saturated, filling up the upper midrange as they fade away.

If you want, send me an unprocessed drum loop and I'll try to make it sound like what I think you are asking for. If I get it right, I'll let you know the exact chain I used, and how to get there even if we don't have the same plugins.

But seriously, savagely overcooking Satin will probably get you a lot of the way there.