In any agile framework, if you want to estimate based on fixed scope, your estimated release dates will vary. If you opt instead for a fixed release date or routine release cadence, then it's scope that varies. Both are legitimate options, but in this case your delivery manager is more right than you are.
All Sprints are Potentially Releasable
Your delivery manager is mostly right. Whether you're following Scrum or some other framework that just happens to use the term "Sprint," an agile iteration should always result in a potentially-releasable product. Scrum in particular makes it very clear that each increment must be releasable:
At the end of a Sprint, the new Increment must be "Done," which means it must be in useable condition and meet the Scrum Team’s definition of "Done"...The increment must be in useable condition regardless of whether the Product Owner decides to release it.
While most agile frameworks don't define a fixed release cadence, neither do they prohibit it. Since each Sprint results in a potentially-shippable increment, there's no reason not to target a three-month release cadence for the project if that adds business value.
A Release Cadence is an Agile Timebox
It's more traditional to see agile release planning estimating a release date based on how many iterations it will likely take to complete a given set of features. However, Lean flow, DevOps culture, and an industry shift towards continuous delivery and continuous deployment have resulted in some projects moving towards a routine release cadence such as the one your delivery manager wants.
As one example, Ubuntu has a biennial release cadence for long-term support releases, and a six month cadence for interim releases. In contrast, the Debian release cycle follows a more traditional model that isn't inherently timeboxed.
Should Releases Follow a Cadence?
A reliable cadence is generally a good thing in a project management framework. However, an iterative delivery cadence and a product release cadence are really two different things. Whether releases should follow a cadence or not is a product-specific business decision, and there's no "one size fits all" answer to that question.