I beg to differ. Your analogy fails to distinguish between the primary roles of Carrick, Matic, Pogba, and Herrera.
Carrick is a deep-lying playmaker. He sits in front of the defence, and uses his vision and passing abilities to open up the opposition.
Matic is United’s number six. He plays in front of the center-backs with the aim of breaking up opposition attacks, and collects loose passes.
Pobga is a powerhouse. He makes shuttling runs on and off the ball to asset his physical presence. He also puts in tackles and makes passes.
Herrera is a box-to-box midfielder. His aim is to make an impact at both ends of the pitch.
In contrast, Mata and Mikhitaryan are both ‘trequartistas’. The trequartista is the link between the midfield and attack.
Griezmann is also a trequartista. United don’t need a third player in that role. In my view, Griezmann would be little more than a marquee signing.
It really depends on how details you want to compare a player.
Most people categorize Matic and Carrick as DM, simple as that. But if you say that Matic and Carrick are "different", then in your case, almost every player on the earth is different with each other. In statistics we call this "cluster analysis", that is, task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense or another) to each other than to those in other groups (clusters). It highly depends on how details you want it to be, thus giving you a different result.
It is not a right or wrong thing, just a different perspective.
For example, Jose clearly brought in Matic as a replacement for Carrick and/or free up Herrera. In the case of Herrera, he was needed to play as DM (you know his best role is B2B CM) last season, the best example was our biggest game of the season in EL final where Herrera played deep, and Fellaini/Pogba both played as CM. Also, as evidenced by the first 4 games of the season (including Super Cup), Matic clearly played as DM, freeing up Pogba and Herrera. So I would say Matic is indeed a replacement for Carrick/Herrera. Matic/Carrick/Herrera's (some games last season) main role is to sit in front of CB, hence they are called DM, it is that simple. Of course, you can go into the very details that they all have different attributes, like Matic is more aggressive than Carrick, has better footwork than Carrick, etc. Then in that case, almost every player is different. It is very hard to compare or categorize then. So I believe majority here would like to say that Matic and Carrick are DM, they are the "same type of players", so that we all could discuss and express our opinion.