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Getting the proportions of ice cream to milk right is the most critical part to having a consistent thick shakes IMO. Use a good dense ice cream (not a light whipped one with lots of air) and use a good WHOLE MILK for thick creamy shakes. But getting the ratio in the cup right is critical. It is less milk than you think you may need–usually less than 1/3rd milk in the cup bottom. So you will need to experiment with the cup size you have to get the right amount of milk to ice cream ratios. Today I see folks use a small scale.. The BEST shakes I get and make come in two cups–the milkshake I want and a small “shot glass” taster of shake that was left over in the metal mixing cup that seems to never make it to the final serving cup due to size-so the point here is to make sure your serving cups can hold whole the ratio cause sometimes the best part is what is left in the mixer cup.. Also–you will want some LONG SPOONS like bartender or mixer spoons to give a small stir on the sides while it mixes–and also know it takes a bit of mixing time (1-2 minutes) to make a good shake in the spindle mixers. Lastly–get a good quality straw to serve with the shake–cheap plastic ones are so so–you want a straw that has a good diameter to allow milkshake liquid and ice cream to be sucked up but not so thin that clogs or it is a huge task.
TIP—make sure your mixer cups are metal–they keep things cold and are easy to quickly clean IMO..and have several (cause as you make you will need to rinse and re-use again in service often and you may have several going at the same time…The link you provided is the right style–but shop around…they do come in pairs or triples–meaning you can get 2-3 shakes going at the same time etc. And if one goes down–you are not screwed til you get a replacement. Also – always have a mixer cup of hot water to clean the spindle after each use too. Have several mixers and metal mixer cups..
As for the shake–get a good WHOLE MILK and a dense ice cream as mentioned–but You can so some enhancements if you wish–whipped cream on top, cherry’s etc…but there are cookie or candy toppings etc–up to you.. For flavors–a dash of vanilla or some fresh vanilla bean into a shake is always nice. For chocolate–a squirt of chocolate syrup is good if it is needed or a tiny pinch of kosher salt in the chocolate before blanding can help to enhance the flavor…but whatever you do make sure you mix well to incorporate. There is a technique to using the spindle mixers to get a well blended shake–you don’t want pockets of milk or cream or whatever unblended or trailing into the top of the shake cause it sat at the bottom. It should always go MILK, ICE CREAM and then any additions–then BLEND. Some make the mistake for example by putting chocolate syrup first so it sits at the bottom and this is wrong.
My two cents–hope this helps (FWIW–worked in a soda-jerk type old fashioned ice cream shop for years when I was in high school long ago–I know my shakes!)
EDIT: fixed link