The quoted text applies only to backgrounds
That text is found in chapter 4: Personality and Background which tells us right from the start that the rules found here most likely only apply to backgrounds.
Furthermore, the text in question states in full (emphasis mine):
Each background gives a character proficiency in two skills. Skills are described in the Using Ability Scores section.
In addition, most backgrounds give a character proficiency with one or more tools. Tools and tool proficiencies are detailed in the Equipment section.
If a character would gain the same proficiency from two different sources, he or she can choose a different proficiency of the same kind (skill or tool) instead.
The previous two paragraphs are specifically about backgrounds so it makes sense to assume the third is as well.
Another reason to believe this is not a general rule is its location. Why would such a rule appear in the section on backgrounds and then not in the rules on "Proficiency Bonus" where one would really expect to find it?
However, the greatest piece of evidence is various features that do let you choose another proficiency if you already have one:
The various Artificer subclasses grant you a tool proficiency, and explicitly state that if are already proficient in that tool you can pick a different tool to gain proficiency in.
If there were a general rule that acquiring the same proficiency again meant you could pick a new one, then there would be no need for this feature to state what it does. The same logic applies to the Samurai Fighter's Elegant Courtier feature and the Gloom Stalker Ranger's Iron Mind feature though both grant saving throw proficiencies, so one could potentially argue they aren't actually covered by the text in chapter 4.
If you'd really like to see a tweet from lead rules designer Jeremy Crawford, that exists too:
The proficiency rule for backgrounds applies when you gain your background at 1st level. It has no relevance outside that context.