I've started to organize my files and for easy sake I'm using my browser for it.
I've come to the point where I want to tag my files and when I select a folder (multiselect + shift click first to last file) and add keywords it looks like things are getting processed, but in the end, no keywords are added to my files.
Can you talk us through the workflow? You can't "select a folder" in LrWeb, you can either select All Photos or you can select an individual album. So which one are you using?
Once you have selected your source, yes you can select multiple images, and then in the blue selection bar you can choose the Keywords option, whereupon you can enter new keywords which will be applied to all selected keywords (or promote a keyword from the "Added to Some" section to the "Added to All" section). Once added (and the change has synced to the cloud) you should see those keywords whenever you view the keyword panel on any of those images in any of the apps in the cloud system (apart from LrClassic if you have synced your Classic catalog). That should be the case irrespective of whether you're viewing an image in All Photos or a containing Album. So I'm a bit confused when you say you can see the tags have been added, but "they do not show up in an Album". Are you still using LrWeb at this point or have you switched to a different app?
In my Album I'm using the multiselect tool, and selecting all files via clicking on the first + shift-clicking the last.
In the blue bar, I select keywords and type the ones I need. I see the processing in the background and once done I click on close. But after doing this action, I check a random file in this album for its tags and they have are not visible.
Now when I do a search for the tag, the file is showing up which means the tags was applied.
Also, when I go to another Album that contains that file and I check the tags, I can see the tags, however, when I go back to my first album and check the tags, the ones I added are still not showing.