Tap the profile picture to expand it, then long-press the image until a menu appears. Select Save Photo (iOS) or Save to phone (Android) to download it directly to your gallery.
Facebook often compresses images to save space, which can lead to a loss in sharpness. If you need original resolution or want to download multiple photos at once, consider these tools: Export a copy of your Facebook information
For a single photo, the official methods are the fastest and safest.
Open the profile photo to full screen, right-click the image, and select "Save image as..." . Alternatively, click the three-dot menu (top-right or bottom-right depending on the view) and select Download .