Oat drinks have significantly less protein than regular milk - that's the first thing you should check if you're just switching over casually. With Oatly and similar products, it's typically around 2g per 100ml, whereas with regular milk it's double or more. So if you don't actually have lactose problems and your digestion works fine, you're basically breaking something that was working well before. The "better for your gut" marketing only really works on people with actual issues - for everyone else it's just a different product, not automatically better.
As for taste, yeah, you'd notice the difference. Oat drinks taste sweet and kind of different, some people think that's great, others see it as artificial. Here you should only care about: what do *you* like? If you normally drink milk and it tastes good to you, there's no reason to switch just because it's sitting on the shelf at Rewe. Stay critical about these trends - just because a product gets fancy advertising doesn't mean it's better for your situation.