Can I just rinse a shaker, solely used for immediate consumption of whey with milk?

So let’s be clear - rinsing with just water won’t remove all of the fats and proteins in your container. It’s definitely not “clean” afterwards.

But even for the bits left in the container food safety guidelines apply, so if you immediately freeze the shaker in between uses, you just have to add up the time the shaker spends in the danger zone and/or above freezing. Whatever value reaches the „no longer safe“ threshold first (two hours for the danger zone above 40°F/4°C, three days in the fridge) gives you the “now it’s really time for a real cleanup” trigger.

Off the cuff, I‘d say that your three day cycle would still be technically safe. Whether it’s really a good idea is up to you. I like my food prep equipment fresh and clean for each use. More frequent cleaning will also serve as extra safety net in case some cleaning cycle was not perfect. And of course in a commercial setting all food handling and cleaning regulations have to be observed independently of what I outlined in this answer.

I agree with Stephie here about the food safety aspect - if you use the freezer, it will be safe. The logistics of it are much more complicated than just washing it though, to the point where I don't think you are doing yourself any favors.

First, you cannot dry it, or your towel will get all fatty. You will have to place a dripping wet shaker in the freezer, so you'll suddenly have to defrost the freezer very frequently. Then you'll get it out for use, and have a frozen shaker - just trying to screw the cap back on might take more time than washing it in the first place. Then, when making your drink, you will freeze your fingers off, and your drink will be very cold (although to be fair, many people like their shakes very cold). And in the end, if you use flavors, they will mix between the different batches. And you have something taking up space in the freezer, which most people find is in too short supply in the first place.

If you decide to use the fridge instead of the freezer, you get a different set of problems. While the fridge doesn't freeze (usually), having excessive amounts of water in it is no fun either and you'll have to soak it up with a towel pretty frequently. Between that, you'll have other items in the fridge getting wet now and then. Also, you will have to keep track of when to wash the shaker - the safe time is 3-5 days, or earlier if you notice spoilage, so you have to be organized enough to know you haven't exceeded it. Also, because it runs counter to best practices, guests who encounter it might judge you negatively.

Yes you can/could (you're the king/queen of your kitchen, no one will come crashing through your door).

But you shouldn't ... make a habit of cleaning it up after each use.

No need to go nuclear, a few drops of cleaning liquid and hot water and a quick wash should be enough.