I believe you are mistaken.
Fighters get two skills from class, rogues get four. Add in background and fighters have four, rogues have six.
Fighter
Core Fighter Traits
Primary Ability Strength or Dexterity
Hit Point Die D10 per Fighter level
Saving Throw Proficiencies
Strength and Constitution
Skill Proficiencies Choose 2: Acrobatics, Animal Handling, Athletics, History, Insight, Intimidation, Persuasion, Perception, or Survival
Core Rogue Traits
Primary Ability Dexterity
Hit Point Die D8 per Rogue level
Saving Throw Proficiencies
Dexterity and Intelligence
Skill Proficiencies Choose 4: Acrobatics, Athletics, Deception, Insight, Intimidation, Investigation, Perception, Persuasion, Sleight of Hand, or Stealth
I am not mistaken, you need to read further down in the PHB. The Rogue must take expertise in two of those skills.
"Level 1: Expertise
You gain Expertise in two of your skill proficiencies of your choice. Slight of Hand and Stealth are recommended if you have proficiency in them."This leaves her with only 4 skills (including 2 from background) that she has normal proficiency in, the same number as the fighter.
When it comes to skill proficiencies in tier 1, the difference in the Fighter and the Rogue is the 2 skills the Rogue has expertise in. After these two skills both of them have the same 4 basic proficiencies out of 16 remaining skills.
I clearly stated that context multiple times above, including right after the verbiage you quoted: "In tier 1 Rogues and Fighters are both proficient in 4 skills. Rogues additionally have expertise in 2 skills."
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