I have 2 questions...
I'm not a licensing expert, but here's my interpretation
1. An asset is licensed for a single client. You can't freely make the asset available internally for anyone else to use on other projects for other clients.
2. no, the client doesn't need to have an Adobe stock subscription to use the asset again. You cannot make use of that license for another client, though.
Thank you for the reply! I do want to try to ask again for clarification. The asset that we (the ad agency) are wanting to license would be on our client's behalf. Eventually, we are wanting to add that asset to our client's asset library. The images in that library are used by that client or for that client only. It could be used for several projects, but always for that one client. Partnering agencies may want to use that image, but for that client only since it is part of the client's asset library. If used for only that one client, is it okay to place that stock image into the client's asset library if we have an Extended License agreement?
Eventually, you should ask a lawyer to see if the use you want to have is in accordance with the licensing terms. If you are considering buying many assets/month, you could also try to check with Adobe, if there is a solution tailored to your situation.
All other discussions you eventually have here, will be speculative and Adobe will not join into this discussion, except to tell you, that you should study the licensing terms.