How to tile windows without going into full screen? Currently, if I want to tile two instances, I press the green button on the top-left and place the instance accordingly.
However, each of the tiled instances always go into full screen mode. For a chrome instance, this is annoying as it means it's not possible to see the tabs.
How do I tile windows without making them go  full screen?
Pressing the green traffic light button would cause the window of most macOS apps to go full-screen.
You can press and hold the Option key when clicking on the green traffic light button to prevent the window from going into full-screen mode.
If you want to tile and align various windows belonging to same of different apps, you can use one of the many 3rd party window and tiling manager apps.
One such app that I use and find very useful is Divvy - Window Manager. It lets you tile windows quickly and efficiently for any macOS app and supports multiple displays too.
P.S.: No affiliation whatsoever with the developer of the app. Just a satisfied customer.