Some of the users do not want the validation message after editing the Standard form. I want to show a message(inside a lightning component) when someone clicks on standard 'Save' button. This should be applicable to both situations 1) When user does Inline Edit and click Save. 2) When User clicks on Edit button and make changes and then click Save.
You can't override standard functionality like that. There's no injection point for you to add your Lightning component over the standard Save button without you overriding the Edit action and reimplementing the whole page, which is an expensive proposition.
The right tool to use to protect data integrity is Validation Rules, alongside marking fields Required as appropriate.