At #4 or larger wire (yours is #2 TW), you are allowed to use colored electrical tape to identify black wires as neutral (white or gray) or ground (green). Note that this has not been done, which is a Code violation, but a very common one.
100A is the correct breaker for #2 TW wire.
Service panel buses actually "zig-zag", like this.
So breaker rows 1 and 3 are on one phase (call it "black") and rows 2 and 4 are on the other phase (call it "red"). Here's a great 25m video on split-phase.
The reason they aren't hooking to the lugs is this is (apparently) a main service panel, and requires a main breaker. If power went to the lugs, then nothing would prevent the service from being loaded to 150A (100A sub + 30A dryer(?) + 20A DFCI on each phase).
This is a type of main breaker called a "Back-feed main breaker". It's a regular breaker supplied in reverse (they don't care, it's AC). Code requires back-fed breakers be bolted or strapped down, so it can't "rock out" like other breakers. If yours is able to "rock out", post a photo of the panel label - we may have an answer to that.
Many service panels are factory-assembled this way. However it's fine to field-assemble a panel like this, and then you end up with a superfluous set of main lugs. (or maybe, not so superfluous.)
Those main lugs
In the 1970s, breakers over 60A were prohibitively expensive. So the "Rule of Six" was written that said a "main breaker" could be several breakers, as long as there were no more than six hand throws. However this design was prone to being overloaded when new loads were added to the house, so it was outlawed as soon as >60A breakers became affordable. So you do need the 100A main breaker, and cannot do without it.
However, here's an interesting question. If the main breaker is 100A, why is a second 100A breaker needed for the subpanel? The answer is, "It's not". Since the main 100A breaker assures subpanel current can't exceed 100A (nominal), the second breaker really is redundant. So how could the subpanel feeder be connected without use of a breaker? The main lugs!
Essentially running the whole panel in back-feed.
That would free up 2 spaces here, which could be used for other things.