Short answer:
Install a PDF reader app to your device. Or find a browser (or other app) which suits your needs.
Long answer:
Android from the start has a concept called Intents, which means one app could use a different app to handle specific task like 'open email app to start a message' or 'play a video'. This also means if a document format has a version 2 / or additional features, the receiver app could be updated to handle it.
The downside is Android devices need to have an app installed capable of reading the file/intent/URI passed to it.
From what I understand iOS only added Universal Links in iOS 9 which could provide similar behavior. Since Apple provides a PDF viewer in their UIWebview/WkWebview I expect any app on iOS to use the existing iOS framework as cross-app interaction had additional requirements.
Given that any additional feature (like displaying PDF) will add size to the original app, in the early days it was better to have a distinct separate app. Didn't need the feature, you didn't need to download anything which could cost valuable data airtime or take up space in your phone and possibly avoid having a duplicated function in different apps. Understandably modern phone have lots of storage space but in the end Android has had a guiding principle via the Intent system to leave the ability to read/play any document (PDF, office file, video, audio) to those apps who can specifically handle those file formats.
Answers to Edit1 and Edit2 by OP:
Edit1 response: "Rooting" won't give you additional features in any browser/app as adding features involves deep changes to the app requiring software development time. If you want a well integrated PDF viewer in a browser app, you'll have to find one. Opera has been suggested. This would be the same with any other messaging/communications app for any particular file or action, i.e. you don't expect it to view spreadsheets or handle geographic mapping. The advantage of Android is that you are free to change which app handles the action and get a better experience rather than be limited to any prebuilt apps (which iOS limits).
Edit2 response: Manufacturers are free to add in their own PDF/Document viewers for their particular flavor of Android. AOSP just provides the environment. PDF at this point is a mix of open and proprietary parts so depending on the document version any PDF rendering engine may encounter difficulties. Additionally PDF is trying to display text/images in a particular way including fonts. This now means that any app embedding a PDF engine will need be aware of security attacks. See my answer to Why has Android made it impossible to install system fonts? on how fonts can pose a security risk. Browsers given their sandbox nature displaying arbitrary HTML and Javascript code are attentive to security, other apps perhaps not so much as it is not a core product feature.
