Both common and pygmy hippos are included as priorities for captive breeding on the IUCN captive breeding recommendations list:
IUCN Red List Captive Breeding Recommendations | Conservation Planning Specialist Group
Ooh, not quite! The IUCN database near-blanket adds a 'recommended for captive breeding' tag to most threatened species, which is just that, a database tag.
At the global level, the IUCN's SSC Hippo Specialist Group, amongst others, develop the foundations for collective taxon-associated conservation objectives: in the case of hippopotamus, the current global focus for
amphibius was the establishment of regional
in-situ management plans - for which there were none. In contrast,
in-situ plans for pygmies are well established, and focus has since shifted to include a more robust
ex-situ strategy. As such, pressure has been put on institutions globally through the mid-2000s to take on pygmies (hence the ballooning number of collections keeping them in the last few decades).
In Europe, which concerns us here, the EAZA's Taxon Advisory Groups (TAGs) use the above global objectives to define the European ex-situ management plans, in increasing order of management level and priority: i) monitor programmes; ii) European studbooks (ESBs); iii) European Endangered Species Programmes (EEPs; equivalent to SSPs with AZA).
Pygmy hippopotamus are currently managed by an EEP. As of 2017, with the publication of the latest evaluation, the captive pygmy population had reached carrying capacity with the ~140ish current holders, and in order to maintain growth rates to reach target population goals (and mitigate some of the captive pop. problems, e.g. sex ratio skew), to quote Steck, B. (2017): "new holders (are) urgently required", and thus remain a priority EAZA species.
The EAZA management plan for common hippopotamus isn't as mature. Unlike the pygmy hippo, they still lack an EEP, remaining a studbook managed species (ESB). We make sure, y'know, they don't inbreed, die out and everything, but there's no planned trajectory towards, for example, defined population targets, and there's comparatively little pressure on institutions to take them on.
The AZA developed an
amphibius SSP in 2014; I suspect EAZA might follow suit and upgrade common hippopotamus management plans in the near-ish future. Until then, as above, they're currently 'not a priority'.