Please, don't overwrite page properties with properties from a specific component (by whatever means). Instead you should write your title component in a way, that it can display the correct value by reading it from that specific sub-component. In that case I would explicitly build logic into the title component Sling model, which looks for another component on that page and reads the correct title from there (preferably using the model class of that component instead of directly reading from the repo).
In general, I consider this not a good pattern. Because the title of the page is defined at a page level, and not on a component level. You should rethink if your properties are located at the right components.
Yes, it is possible, but I'm not sure if a component would be the best way to perform this task.
For example, in order to write the custom code for updating the page properties, you would most likely create a Java Sling Model for the component.
In this Sling Model, you could get the current page, check if the title matches the value mapped from the configuration of the component, and if they don't match, update the page properties.
The reason why you likely wouldn't want to do this with a component is because that Sling Model will do that check every time it is rendered on a page, and you would basically only want to run that check once since it would just be an update that occurs once when the component is authored or edited.
Not sure what your exact use case is, but editing the page properties to have a different title isn't that difficult from an authoring standpoint, and I don't really see the ability of being able to update the page title via a component creating a better authoring experience.
Perhaps training your authors on to how to easily update the page properties to have a different title might be a better route than having to do a custom implementation on a component.