I disagree with the previous posters and think there are some AE/BE differences here.
An exhibit refers to the presentation of animals to the public for them to view, though in BE the word exhibit is rarely used anymore (sounds not very PC). Like German, these are now more often referred to generally as attractions or also collections.
(AE -> The tiger exhibit is open 09:00-18:00
BE -> The tigers can be viewed 09:00-18:00)
An enclosure is what the animals live in.
(BE/AE -> The tiger enclosure is quite secure)
In German Zoos/Tierparks I observe the following...
The animals are kept in 'Gehegen' or 'Anlagen'. Specific examples of which are: Streichelgehege (for donkeys as example), Freigehege (for deer as example), usw....
What the Zoo or Tierpark has are not normally described as exhibits/Ausstellungen of animals but 'Attraktionen'.
When they do have exhibits/Austtelungen... the Austellungen are often lumped together with Veranstaltungen and refer to specific or special events, such as an artists collection of paintings of Zoo animals or a special bird-of-prey display not given every day. These are usually designed to raise extra money for the Zoo. These are exhibitions in AE/BE.
Refer to some Zoo/Tierpark sites if you'd like more evidence to support my impressions.