die Freigabe
approval
Official permission or authorization to use, publish, sell, or begin something.
Der Softwareentwickler wartet auf die Freigabe des neuen Updates.
The software developer is waiting for approval of the new update.

approval / release
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approval
Official permission or authorization to use, publish, sell, or begin something.
Der Softwareentwickler wartet auf die Freigabe des neuen Updates.
The software developer is waiting for approval of the new update.

'Freigabe' (approval/release) is the noun form of the verb 'freigeben' (to release/approve). It is made up of 'frei' (free) and 'Gabe' (giving) and describes the act or result of 'frei zu geben' (to release), that is, to permit something or make it accessible.
Die Freigabe der neuen Softwareversion erfolgt morgen.
The release of the new software version will take place tomorrow.
Very often, a genitive or the preposition 'für' (for) is used to specify exactly what is being approved or released. This is typical in formal and official contexts.
Die Freigabe des Budgets steht noch aus.
The approval of the budget is still pending.
In the context of permissions, 'Freigabe' (approval) is often used with the verbs 'erteilen' (to grant), 'erhalten' (to receive), or 'benötigen' (to need). These combinations are very common in professional and administrative life.
Der Projektleiter muss die Freigabe für die nächste Phase erteilen.
The project manager has to grant approval for the next phase.
When it refers to lifting a closure, you will usually find 'Freigabe' (reopening/release) in connection with traffic routes. Typical nouns are 'Autobahn' (highway), 'Straße' (road), 'Brücke' (bridge), or 'Strecke' (route/section).
Die Polizei kündigte die Freigabe der Brücke für den Abend an.
The police announced the reopening of the bridge for the evening.
In IT, 'Freigabe' (sharing/access) refers to access permission for digital resources. Signal words for this meaning are 'Datei' (file), 'Ordner' (folder), 'Netzwerk' (network), or 'Link' (link).
Kannst du mir bitte die Freigabe für diesen Ordner erteilen?
Could you please give me access to this folder?
approval / consent
'die Zustimmung' and 'die Freigabe' can both have to do with permission. 'Zustimmung' comes from a person or body that agrees to something. 'Freigabe' is often official permission that allows something to be used, published, sold, or started. In addition, 'Freigabe' can mean that a block or closure is lifted, for example with a road or a file. The words can be confused when it comes to approvals. As a practical rule: for personal or general consent, you are more likely to say 'Zustimmung'; for an official launch, access, or technical access, 'Freigabe' is more likely.

Die Freigabe der neuen App erfolgt erst nach dem letzten Sicherheitstest.
Approval for the new app will only be given after the final security test.
Nach dem Gespräch mit der Leitung haben wir die Zustimmung für den Ausflug bekommen.
After talking with management, we got approval for the trip.