In order to get the Collector achievement you'll need to get every items of the game. This includes all kinds of usable item, equipment and more. You will even have to get all Valuable items and as for the Knowledge section I don't think it is required but since I'm not sure about that I will include all info about those as well.

What I recommend:

• Check out the few missables and be sure you get them during your playthrough.
  
• Complete every Chapter and SideQuest as some item will only be available after completing certain events. You even have to beat Galdera (final boss) to get the Spurning Ribbon.
  
• Get every chest in all location.
  
• Buy/Steal and Inquire/Scrutinize every NPC, some items will only be available that way.
  
• Visit all shops to buy everything else you don't already have.
  
• Once all of the above is done, check out with my listing below (at the end of my guide) what you're missing. There will have a few items you will need to Challenge a specific NPC. Remember that challenging doesn't make the item drops all the time, you may have to challenge more than 1 time before what you are looking for drops. Other than that there's a few items you'll get by stealing or defeating specific enemies. Again those will have notes on how to get them, there's only a few of them.
  
• If you have every regular item and you still don't have the achievement you probably missed a Valuable item. In that case look out the Valuables tab in my listing and revisit every NPC holding those. Even though the sidequest may be completed with alternate solution you'll still be able to get all those valuables after the quest is completed except for the Original Tome of course.
  
• If you still don't have the achievement, check out the Knowledge tab. I'm not entirely sure if they are required or not for the achievement but you have the info anyway.

Here are the items that are missables in the game:

• Obsidian Garb: During Primrose chapter 3, steal the Obsidian Officer to get it. Upon defeat you'll have a 30% additional chance to get the armor. You can get up to 4 if you're lucky.
  
• Original Tome (Valuables): After defeating Lucia (Cyrus chapter 4) you'll need to scrutinize 3 pieces of Knowledge:
  • Trial of the Twelve, Volume VII (right bookshelf)
    
  • Forbidden Gold (left bookshelf)
    
  • From the Far Reaches of Hell (book at Lucia's body)
    Be sure you get "From the Far Reaches of Hell" last else the Original Tome won't appear in your inventory. I would guess this is a bug but I know for sure this one is required to earn your achievement.

• Joshua's Weak Points (Knowledge)
  During Olberic Chapter 2, be sure you Scrutinize/Inquire Joshua before you fight him.
  
• Archibold's Weak Points (Knowledge)
  During Olberic Chapter 2, be sure you Scrutinize/Inquire Archibold before you fight him.
  
• Gustav's Weak Points (Knowledge)
  During Olberic Chapter 2, be sure you Scrutinize/Inquire Gustav before you fight him.
  
• Tidings of Elderly Friends (Knowledge)
  During Alfyn Chapter 3, you'll have to Inquire a Townsperson for "Tidings of Elderly Friends". It's kinda hard to miss, you should see him right away upon exiting the tavern, but you can totally skip him if you go Inquire Demure Grandma right away. Thanks to Weichichi for this one.

I got a report from VictorWonder and multiple other sources saying they earned this achievement while missing those knowledge about the 3 weakpoints from Joshua, Archibold and Gustav.
Thanks to Soulcast for letting me know "Tidings of Elderly Knowledge" is also not required for the achievement.
While I don't know for sure if other Knowledges are required or not to complete the achievement, I know for sure that Valuables are. Anyway try to get everything, this way you'll be sure to secure your achievement.
The Obsidian Garb has also been reported as not required for the achievement.
So in the end, The Original Tome is the only missable required for the achievement.

It is not required to keep all item in inventory. You can sell anything you want for as long you held it at one time it will count, however it is way easier to keep track of everything if you hold onto all your items in order to compare with my checklist.

Sidequests tips, because most Valuables items are linked to those:

• If you need help with sidequests there are many places to go but I like using the following web page because they are categorized by town, so it's easy to check your quest log what you are missing as you just need to count how many quest from 3 town and compare to your log (by region):
  https://www.neoseeker.com/octopath-traveler/
  
• Each Regions (Coastlands, Highlands, Sunlands etc.) will host a NPCs with a sidequests that has 3 parts. Part 1 will always be on Tier 1 town and the following part 2 and 3 will be on the same region but upgrading to the next town. So for example, before doing Sir Miles, Servant of the Flame (III), you'll have to do part 1 and 2 and on my listing you know that part 3 is the Frostlands (Northreach) and because of that you can expect to see him on part 1 in Flamegrace and part 2 in Stillsnow...
  There are only 3 exceptions to this point:
  • Zeph and Mercedes (I) (II): Part 1 in Clearbrook, Part 2 in Atlasdam
    
  • In search of Father (I) (II): Part 1 in Noblecourt, Part 2 in Moonstruck Coast
    
  • Daughter of the Dark God (I) (II): Part 1 in Bolderfall, Part 2 in Saintsbridge

Thanks to Jabberwork for providing his Valuables listing of which I could compared to mine and double check my results.
https://docs.google.com/spreadsheets/d/1DI-pv5CyoqU0L5P8tVGBzslC9N13KM-w-mAK5331Gs8/edit?usp=sharing

Happy item hunting! :)