Game shifts to edge of screen when clicking Play button in Unity

Try opening the Window menu and choosing Layouts –> Default. That should change your window to look something like this:

Default Window Layout

(You can also try the other standard layouts to see if any of them behave better for you)

Click the Game tab indicated by the red arrow in the screengrab above, then click the Maximize on Play button:

Maximize on Play Button

With this enabled, clicking the Play button should make the game expand to fill your whole Editor window.

If it’s still not behaving correctly for you, editing your question to include a screenshot of what you’re seeing may give us some clues for what to try next.