Greetings saikat_75,
Thanks for posting in the Apple Support Communities!
Managing your Apple ID profile picture can be done with different devices, so you'll want to be sure you're up-to-date with the software version on the device you're using. While this may not be the direct cause, it's best to remain updated to see if that's related to what's occurring.
To change the photo via iCloud.com, use the steps below to see if that will allow the change to save. You can find additional information here: Change your Apple ID photo on iCloud.com
1. On iCloud.com, go to Account Settings, then click the photo next to your name.
2. Do one of the following:
• Drag an image file onto the photo.
• Click Choose Photo, then follow the onscreen instructions.
If that isn't helpful, Contact Apple Support directly to check into this in more detail.
Take care!