Human-engineered, canine combinations are known as hybrids or "designer dogs"...I think a better name for these canine combinations would be "portmanteau dogs."
In linguistics, portmanteau words are those that are formed by combining the sounds and meanings of two different words; they are considered a linguistic blend. A canine blend combines the sounds (of breed names) and meanings (of characteristics) of different breeds.
Here are some examples:
Bernedoodle = Bernese Mountain Dog + Poodle
Bullmation = Bulldog + Dalmatian
Chipins = Chihuahua + Pinscher
Labrottie = Labrador Retriever + Rottweiler
Maltepoo = Maltese + Poodle
So maybe if Obama goes with a Labradoodle he could name it Portmanteau and call it Port for short.
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Obama did go with the Portuguese Water Dog and his name is Bo. Even though he didn't choose the "portmanteau dog" he still could have named the new dog Port for Portuguese Water Dog.