Definition like in this wiki article. Basically animals that don't need any special training and are just there to comfort their owner with their presence.
The best I could come up with (which happens to also be the result of Google translate) is "Tier zur emotionalen Unterstützung". Sounds way too clunky to me though. Are there any better options?
There are already some excellent answers and suggestions given, but per request I'm posting my own.
First, the idea of an emotional support animal (ESA) seems to have legal meaning in the USA only. The Americans with Disabilities Act of 1990 prohibits discrimination against people with disabilities, both mental and physical. This is similar to the Civil Rights Act of 1964 which prohibits discrimination because of race, religion, gender, national origin. Service animals, notably guide dogs, have been in use around the world for long time, and as a logical result, the ADA extends protection to these animals and exempts them from restrictions placed on "pets". The idea of service animal has been extended to include animals used to support with certain mental disabilities such as Autism, PTSD (PTBS), and acute anxiety, and these have come to be known as emotional support animals, especially when they require no specific training for this role. In recent years the lack of a specific definition and the fact that no training is necessary for these animals has lead to airlines accusing people of gaming the system and using the exemptions to carry exotic or even dangerous animals on planes, and the Trump administration put restrictions on what qualifies as an ESA.
A related idea is that of a therapy animal, meaning an animal which interacts with humans as a form of medical treatment. There doesn't seem to be a clear line between a therapy animal and an ESA; many service animals provide emotional support in addition to enabling mobility. But there does seem to be a difference, with animals used purely for treatment at one end of the spectrum and animals which serve purely to help people deal with a disability at the other end.
The lengthy preamble was to show that the whole idea of an ESA may be rather alien to German speakers, and that German speakers may be skeptical of the reasons for such a role for animals. (Many Americans are skeptical as well, but at least they are usually aware of the concept.) In general, you can create words for new concepts in German just by stringing words together. Here are some terms I found on German Wikipedia:
As you can see, allowing for variations of phrasing, the German terms are just words strung together to produce the desired idea. For example Blindenführhund translates literally to "blind lead dog".
guidot suggested Schoßhund which translates to "lap dog". Certainly such a dog would provide emotional support, but to me it would considered a pet rather than a service animal, however loosely defined. Similarly Seelenhund was suggested by Paul Frost, but I'm not sure that the term distinguishes between an animal that simply makes you feel better and an animal that you need to have with you to avoid having a panic attack when an airplane takes off. Note also that the site linked in LаngLаngС's answer does not translate the English term at all. (Such phrases strike me as Denglish, but I'm not in a position to criticize.)
So, I think the upshot is that you're free to string together whatever combination of words you want if it conveys the meaning you have in mind. In the comments I suggested Emotionalesassistenztier since it's basically a literal translation of the English, or Therapietier since it's already used in Wikipedia, though I'm not certain the latter term would have exactly the same meaning as "emotional support animal".
The English original is already quite clunky?
But as the concept catches on slowly in the German speaking world, a direct import for this kind of Assistenzhund* is usually explained as:
Emotional Support Animal / Dog
Ein Emotional Support Dog (ESD = Unterstützerhund bei emotionalen Unsicherheiten) unterstützt seinen Menschen im Alltag in belastenden Situationen. Im Unterschied zum PTBS Assistenzhund lernt der ESD häufig keine „klassischen“ Assistenzleistungen.
But as the laws concerning these animals (i.e. dogs) were just very recently changed in Germany:
Mit in Kraft treten des neuen ASSISTENZHUNDEGESETZES (§§12e ff BGG) ÄNDERT sich für den ESA Dog und seinen Menschen einiges. So hat der ESA Dog, auch schon in der Vergangenheit meist nur geduldet KEINE Zutrittsrechte, Mitnahmerechte oder sonstige Sonderrechte! Ein ESA Dog ist, rein rechtlich ein gut ausgebildeter Familienhund, der „zufällig“ bei einem Menschen mit emotionalem oder psychischen Unterstützungsbedarf lebt.
So legally, these animals are defined as 'merely well-trained', and when regarded as conforming to these criteria of "emotional support animal", then the direct English wording is used, for convenience jsut as the abbrevation 'ESA dog', or 'ESAhunde', or 'ESD'. As these are uncommon as of now, there seems to be no solution available other than explaining what this concept is supposed to mean.
*: Note that others firmly deny that ESDs would be any form of Assistenzhunde…, but note how this as well is hanging on legal definitions:
ESAHunde helfen ihrem Besitzer auch, haben aber nicht so viele Rechte wie Assistenzhunde.
I suggest
Schoßhund
even when it is defined in DWDS more as recipient of extended caressing. I would consider it as implied, that there is some emotional benefit on the other side as well. The noun is quite focused on a dog, however, and I see no more generic noun with the same flavour.
It is also implied, that the dog is smallish, so it can conveniently reside on the lap.
A common set phrase for "emotional support" in German is "emotionaler Beistand". Building on this, you could maybe use the phrase "Beistandshund" or "Beistandstier".
This is probably not a well-established word, but a quick search shows that it is at least occasionally used with the intended meaning. However, there are also uses in the more general sense of "support animal" (synonymous to "Assistenztier"). So it might be necessary to specify this further when first introducing it: "... ein Beistandstier zur emotionalen Unterstützung der Besitzerin/des Besitzers"
Although you ask for a German version of "ESA" which seems to have a therapeutical context, I suggest
Seelenhund (soul dog).
This is a dog being in harmony with a person's soul and making feel human beings better. It is not an official term appearing in legal documents, but I think it covers the idea.
