Rulings, Not Rules
Jeremy Crawford, the lead game designer, called the rules "intentionally silent on these corner cases":
Wild Shape can introduce wild situations. What happens when someone swallows a druid in a Tiny form? Is a druid fecund in beast form? The rules are intentionally silent on these corner cases, leaving adjudication to DMs. As always, I say go with what's best for your story.
The particular ruling (the druid can shapeshift into a specific animal) can lead to a very satisfactory in-game situation. Or it can be utterly boring and devastating, depending on the plot. It is the DM's job to make the right decision.
Aside from the combat, there will always be huge difference between classes' features, so you can't compare (or "balance") them. Some features will be much more useful than another ones in certain situations.
Following the rules-as-written as strict as possible won't help here. In the end of the day, making the game fun an engaging is not about the rules. For instance, if your game is all about picking locks and disabling traps, and you have only one rogue in the party, (s)he inevitably steals the spotlight. You, the DM, have to balance these things by your self — how exactly do class features work in order to not to spoil the fun.
You are the DM and you want to decide, how does magic (the Wild Shape, in particular) work in your world. For this job, what things should you consider in the first place — your own story, the fact if your players have fun, common sense, or nitpicking these minor semantic details in the rules (which are concise and not detailed enough)?
The Adventurer's League Guide describes the role of the DM the similar way:
As the Dungeon Master, the most important aspect of your role is facilitating the enjoyment of the game for the players. You help guide the narrative and bring the words on the pages of the adventure to life. The outcome of a fun game session often creates stories that live well beyond the play experience at the table. Always follow this golden rule when you DM for a group: Make decisions and adjudications that enhance the fun of the adventure when possible.
Wild Shape description is open-ended
PHB gives only basic restrictons of the Wild Shape:
Starting at 2nd level, you can use your action to magically assume the shape of a beast that you have seen before.
Your druid level determines the beasts you can transform into, as shown in the Beast Shapes table.
The only limitations it describes are the creature type, its CR and its flying/swimming speed — pure balancing ones, a sound base for DMs to build their own adventures. As a DM, you are free to apply all the necessary restrictions — the creature size, type, features or appearance. It would be reasonable to discuss this with the player beforehand, say, prepare a list of their wild shapes.
You don't have to though — if you are happy with the default restrictions, just say that all other things are allowed (dinosaurs included), unless it spoils the fun.
Polymorph (4th level) is limited in what it can turn things into.
From the spell text (second paragraph):
The new form can be any beast whose challenge rating is equal to or
less than the target’s (or the target’s level, if it doesn’t have a
challenge rating).
A beast is a particular creature type. The creature type is listed in the stat block of a creature. Compare adult red dragon (dragon) with allosaurus(beast).
Beasts are nonhumanoid creatures that are a natural part of the fantasy ecology. Some of them have magical powers, but most are unintelligent and lack any society or language. Beasts include all varieties of ordinary animals, dinosaurs, and giant versions of animals. (Basic Rules, p. 110).
Other types include fiend, fey, dragon, aberration, celestial, construct, monstrosity, ooze, and humanoid. Unless the creature form chosen is a beast (allosaurus, brown bear, etc) the 4th level polymorph spell won't turn the PC into a dragon. As far as the game is concerned, a dragon is not a beast; a dragon is its own type of creature.
Dragons are large reptilian creatures of ancient origin and tremendous
power. True dragons, including the good metallic dragons and the evil
chromatic dragons, are highly intelligent and have innate magic. Also
in this category are creatures distantly related to true dragons, but
less powerful, less intelligent, and less magical, such as wyverns and
pseudodragons. (Basic Rules, p. 110).
The 9th level spell, True Polymorph, has no such restriction.
Creature into Creature. If you turn a creature into another kind of creature, the new form can be any kind you choose whose challenge rating is equal to or less than the target’s (or its level, if the target doesn’t have a challenge rating).
Your friend may be confusing the two spells, or may not see the distinction between the two creature types. On a humorous note, @NathanS suggests that the most powerful choices for polymorph (4th level) are Godzilla (T-Rex, CR 8, beast) or King Kong (Giant Ape, CR 7, beast). How does the character, or the player, feel about being polymorphed into one of those magnificent beasts? 😎 The PC would need to be level 8 or 7 (respectively) for that to work.
Best Answer
No because Unique/Named creatures are not the "average example"
As @Crazjtk correctly pointed out the examples you gave, Titivilus, Yan-C-Bin, Kiril Stoyanovich, and Ahmaergo, are not unique creatures but rather exemplary examples of a particular kind of creature. This is easy to see in the cases of Kiril Stoyanovich and Ahmaergo who are a werewolf and a Dwarf respectively.
It is harder to tell in the cases of Titivilus and Yan-C-Bin however because we do not know what the "average example" of their type of creature is. We don't actually need to know what an average Titivilus type creature is just that Titivilus is not average. I believe that we can conclude that any named creature is not the average-form of a creature type. If a player encountered Griphook the goblin, whose stats were exactly those of a standard goblin, the player could not shapechange into Griphook only an average goblin. That the two are identical is incedental.
We do not know from the books how the DM defined the origin and history of Titivillus. It could be that in your game's universe Titivillus is an average example of his type of creature, be it because others are all like him or because he is unique. Even if that were the case a player could only turn into Titivilus's type of creature not Titivillus.
Since there is no provided stat block for an average Titivilus-type-creature it becomes up to the DM to define one. A kindly DM might determine that Titivilus is entirely representative of his kind while a DM that resents being made to make up a new monster might declare Titivilus to be nothing more than a particularly empowered Lemure. This isn't a danger with creatures that have defined "average" forms but whenever the average form is not provided only the DMs imagination can fill in the blank.