What does time of year (Early, Mid, Late) and year in About This Mac dialogue mean?
The label indicates the time of the year and the year the particular Mac was announced and released for sale by Apple. The label is commonly used to identify the generation of the Mac model (just like various iPhones are commonly differentiated by their marketing name such as iPhone 7, iPhone 8, iPhone XS etc.)

It does not represent the date of production. Since it is a model identifier, the label remains the same for all the Macs produced till an update/upgrade to the model is announced. The nomenclature generally used is Early, Mid or Late depending on the time of the year the model was released appended with the year of release.

For e.g. the last model of Mac mini released by Apple is labelled Late 2018 as it was released in November 2018. The model it superseded is Late 2014 as it was originally released in October 2014. All Mac minis produced during the four year period were labelled Late 2014 and saw no updates/upgrades from Apple whatsoever.


  I want to buy MacBook Pro that would display Early 2019. If I buy a brand new MacBook Pro from Apple Store (or order it online from Apple website) let's say in February 2019, will it show Early 2019?
  
  What does that year label actually display?


The last release of MacBook Pro was in July 2018 and they are labelled Mid 2018. The one you'll buy in January will be labelled Mid 2018 irrespective of the place you buy it from (unless you end up purchasing an older model, or unless Apple releases a new one in January which will be dubbed Early 2019). Month is never shown as part of the label, so there's will be no February 2019 MacBook Pro.

Now, if you are interested in determining the actual date of manufacturing of your unit, there are numerous ways. The easiest one being using a popular 3rd party utility called coconutBattery. The app is primarily used to determine battery health (charge cycle count), but it also shows the manufacturing date of the machine along with the model identifier.