In No Man's Sky, completing milestones gives you trophies/achievements, but is there any practical purpose to them in-game? Is there a milestone rank required to complete the game?
The only use for these milestones is for finishing the game via the Atlas Path. In order to achieve the Atlas Path ending, you will need to be rank Magellan or higher, which means you will need a cumulative journey milestone rank of 50+, as described in the Atlas Path guide here.
Hit at least 50 journey milestones - you'll need the title of "MAGELLAN" to fully complete the Atlas Path, although each Atlas Interface you commune with will require another rank up in your milestone achievements to continue 'submitting' to it's will.
Also, Polo will only give you a blueprint (including the blueprint for the Atlas Pass V1) if you have reached certain milestones, as described here.
Polo will always offer you a blueprint, Atlaspass V1 on the first encounter, which greatly helps you to upgrade your exosuit's capacity (every starbase has a door that is locked and features an upgrade). In order to get the blueprint from him, you must possess data. First it's learning a few alien words, next meeting aliens, then having earned so many credits, and number of pirates killed.
Other than this, there is no practical use for these milestones.
It's worth noting that it is impossible to beat the game without achieving a single milestone, however, as warping to the center of the galaxy would net you a few milestones for number of warps along the way, even without following the Atlas Path.