A common misconception about Agile is that there is only one framework for Agile. There are many different flavors of Agile out there. Scrum, Scrum-ban, Lean, Lean Kanban, XP, Mob Development, SAFe, LeSS, and our favorite Practical Hybrid. Check out the Agile Practice Guide from Agile Alliance: https://www.agilealliance.org/agile-practice-guide/
Every organization has its own unique culture, and every product has its own specific needs. One size fits all Agile doesn't work. The secret? Understand the many flavors of Agile and pick the parts of those Agile flavors that will work best for your team's culture and your product.
Agile frameworks can vary widely between organizations and even teams within the same organization with so many different approaches available.
Another factor in how organizations approach Agile is based on regulatory, legal compliance, required audits, government approvals, and the level of risk an organization is willing to take when developing or modifying a product. A web application product is far more willing to take risks than a mobile banking application. Nobody wants their checking account balance messed up. A grocery store application doesn't have the same compliance concerns as a medical device application. There won't be any calls to 911 about the shopping cart not working. If your grocery store cart gets screwed up, it won't cause physical harm, although it will undoubtedly annoy you.
In a nutshell, our advice is to roll your own. Learn the different Agile frameworks out there, pick one and use it as a starting point. Add techniques from other frameworks that make sense and show they contribute to the team's velocity. Remove techniques that are slowing the team down or not adding value. Think of Agile as a process. Fit that process into your organization to make it as efficient as possible. It might not be textbook Agile, but it will be the Agile that works most effectively for your team.