I never claimed they would be the exact same character... that's a strawman that you seem to be setting up. My concerns are centered on his main niche. What is the difference in core rogue skills between a rogue and a rogue/fighter with a difference of say 2-3 levels in the rogue class? I realize the classes and subclasses give out individual abilities but can a fighter/rogue pick locks, use sleight of hand, sneak, etc. just as well as a pure rogue... if so I'd consider that stepping on his toes.
EDIT: I would be interested in hearing exactly what these differences are in play however... would you care to go into some detail?
A lot of it does depend on what you define as a rogue's 'core features'. Let's go ahead and assume it is the following:
1) Finding Traps (Perception / Investigation)
2) Scouting (Perception / Stealth)
3) Disarming Traps and Opening Locks (Thieves' Tools)
You could also make an argument for social skills (Deception/Persuasion), or agile exploration (Athletics/Acrobatics), or skillful manipulation (Sleight of Hand) - but for now, let's go with the scout/trap role as the primary niche. Let's see how a pure Thief compares to a Fighter/Thief at certain levels.
Level 2: A Fighter 1 / Rogue 1 can get Deception and Stealth from the Criminal Background, Acrobatics and Perception from Fighter, and Investigation and Thieves' Tools from multiclassing into rogue. He can also get Expertise with Stealth and Thieves' Tools. The Level 2 Rogue can get all those skills from his class, plus has room to get two more from a Background. So at this level, the Fighter is indeed about the same on the primary niche, but the Rogue has a bit more versatility in other areas. The Rogue also has Cunning Action, which likely makes the Rogue a superior scout with how it helps his ability to hide and move around the field.
Level 4: The Fighter 2 / Rogue 2 now also has Cunning Action, catching up to the Rogue in that regards. However, the Rogue 4 is now in the Thief archetype, giving them Second Story work (providing significant scouting ability via climbing and jumping), and Fast Hands (letting them use their Cunning Action for picking locks or disarming traps.) So the Rogue gains a big boost in their ability to deal with such obstacles when time is of the essence.
Level 6: The Fighter 3 / Rogue 3 now is also a Thief, and gains Second Story Work and Fast Hands. However, the Rogue 6 now gets two more skills with Expertise (say, Perception and Investigation), making them again the best at the total skill package.
Level 12: The Fighter 6 / Rogue 6 now also has his second set of Expertise options, and finally thinks he will catch up to the rogue with his skills. However... the Rogue got Supreme Sneak at 9th level from the Thief archetype (making him excellent at Stealth checks), and got Reliable Talent at 11th level (where he can't roll below a 10 on trained skills.) The Fighter/Rogue is certainly *good* at being a scout and dealing with traps, but the pure Rogue remains the *best*.
Now, with or without multiclassing, the truth is that you can make a competent scout/trapfinder with any class in 5E via the right backgrounds and ability scores. But I'd say Rogue certainly lets you do it rather effortlessly, and while getting quite a few useful features that really supports that approach. For me, that's kinda the best of both worlds - the ability to cover a variety of roles without being 'locked in' to bringing along a specific class... but also feeling, when that class is present, that they have something special they still bring to the table in that regard.