Are these special transistors or diodes?
The big difference is that the transistors and diodes are matched because of their monolithic construction which is difficult to impossible to duplicate with discrete parts. Adding emitter degeneration can make up for some of this but introduces other compromises and it will be very difficult to get the same precision as the original parts.
One major problem is that PNP transistors built on an NPN process have very high base-emitter breakdown voltage which parts take advantage of to provide a high differential input voltage range for differential pairs. Duplicating this without these PNP transistors requires a different circuit and again compromises precision.
I was going to try and make something more complex like a LM358(not sure the name it has three leads: heat sink trans. package: to92 maybe? and will regulate voltages) but it has funky transistors with 3,4 or 5 /collectors/ emitters that seem like you couldn't replicate with a standard 2n2222 or "regular" generic transistors parts kits come with.
Multiple collector and emitter transistors can be duplicated with multiple discrete devices but the matching problem mentioned above prevents doing this easily. Of greater concern is that most schematics are either simplified or just equivalent to the actual part and leave out important features. For instance most schematics of the LM358/LM324 do not show the transconductance reduction in the input differential pair, current mirrors, and biasing.
Do they make any kits where you learn a simple IC by making it transistor counterpart?
These days they use SPICE but in the past, special analog ICs were made which had the various integrated matched transistors for prototyping.
So what you are proposing can be done with discrete parts but it requires more than duplicating the circuit as shown.