truth
The truth is you cant get it to sound exactly the same. But, if you listen very closely how each instrument sounds (ambience, panning, frequency content, function within the whole, etc) you can narrow down the traits by confirming what they are not. This will help you get closer to your understanding of why you like certain types of sounds over others. You can also notice subtle details that contribute to the overall perception of the sound in things like, that in the Monkees track some instruments quickly fade out after they are heard at the end of the verse or chorus etc. Probably someone just pulling down the fader at the end of a guitar part etc.
Basically all you can do is examine the sounds and figure out what you like about them and then chase that same vibe in your own recordings. You wont ever be able to get the exact same sound, but if you loved how "honest" a recording sounded, youd be better off figuring out what traits make a recording sound "honest" to you, and then try to make your equipment produce that sound. This way, you may never get the exact sound of a Monkees tune, but at least you will have learned how to get an "honest" (or whatever vibe it is) sound out of your own equipment.
BTW, usually a great song in the old classic "pop" style prioritizes melody, vocal performance, and lyrics first. They are the most important parts and all of them should be focused on, finished, and sounding entertaining on their own without any help from production style. If you have those things then you can move on to the next important things like solid, expressive drumming and bass playing that moves with the kick drum and is not just a low register guitar part being passed off as a bassline. At that point, if everything is working for you, you are practically done. Only a small amount of flavoring is added to those old pop style tunes. Everything follows the list of priorities starting with vocal melody and lyrics, as well as composition/structure. Keep the vocals up front and the lyrics should be amusing or interesting one after another, no lulls. Always go simpler and simpler if you can. Thats how those old songs are.
If you have a solid composition and you record it decently and then balance the sounds by mostly lowering volumes and cutting eq (rather than boosting eq and raising volumes to solve mix problems), you will end up with something that impresses even if it isnt a carbon copy of the Monkees recorded sound.
While we are posting youtubes of the classics, here is one of my favorite ones:
https://www.youtube.com/watch?v=QMtGnVVpdO4