[RPG] Can the Dragonborn Druid use their breath weapon in wild shape

Like most other answers, I would say yes, two shapeshifts in a combat is legit; no, you did not make a mistake by allowing your player to wolf out and; no, you shouldn't try to claw back the ability.

One thing I would add to the previous answers is that this issue will almost certainly balance itself out fairly soon. Yes, a CR 1 creature in a level 2 party is huge. But level 2 ends quickly.

• At level 3, most classes get their archetype paths with cool powers and primary casters get level 2 spells. The wolf will be the same wolf it was before.

• At level 4, a CR 1 creature is considered a medium encounter for a single party member. Everyone will be pumping their primary stat or taking a feat. Wolves don't get much out of Wisdom, nor is there much overlap between the feats that help them and those that work for druids.

• At level 5, all classes see a major power spike. Cool abilities, level 3 spells, proficiency bonus goes up to +3. The party might also be finding minor, but interesting, magical items. While your Druid is kicking it lupine style, they won't get to enjoy any of that.

• By level 6, the Druid can finally upgrade to a CR 2 creature! Of course, at 450 XP, a CR 2 is less than a medium difficulty solo encounter for a character that level.

The dire wolf will remain a CR 1 creature while the world around it scales up. On top of that, if your player wants to jump straight into wolf form every encounter, then every encounter is going to play exactly the same for them, because a wolf only gets the one basic attack.

I'd say, let your player have their fun. Everyone should get a chance to feel like they've beaten the system once in a while. It doesn't seem to be worth the trouble of taking that away from them if the whole situation will simply resolve itself in time.

I'd honestly be more worried that a character with a d8 hit die has 28 hitpoints at level 2.

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.