Yes. This is how all of my freelance work is conducted.
Disregard anything anyone has told you.
The area that covers this is defined legally as a work for hire:
Owner of the Copyright in a Work Made for Hire
If a work is made for hire, the employer or other person for whom the work was prepared is the initial owner of the copyright unless both parties involved have signed a written agreement to the contrary.
The problem is, that few people ever make or write these agreements. But I can guarantee you, it is assumed. And you will not have much luck a) asking for extra money to make it so, or b) getting any agreement to allow you to use works you have created for others outside of those creative projects.
The general “silent” agreement between freelancers and their employers is this work for hire agreement. And usually when you work with a very strict organization they will have to sign a contract to this effect.
It does not matter if you created the concept or not. It is work-for-hire. And no one. I guarantee you, no one, is going to pay extra for this. Because this is the arrangement already assumed.
You can however insist that they put that in writing. If you are looking out for their best interest…lol
If they have not required a contract, technically maybe you could do it. But is this how you want to gain your reputation and run your business? I don’t think so.
Your best practice would be to get them to put it in writing and honor the agreement, build good business relations.
And sometimes clients will ask for an NDA and require you not put something in your portfolio for a period of time to allow them to market it properly first.