Term for words such as "dognitary"

It is a portmanteau, a word formed by merging two other words. Smog (smoke + fog) is a well-recognized example of portmanteau.

That is, if dognitary itself were a recognized word listed in the dictionary, it would be portmanteau.

But, since dognitary is not in the dictionary yet (but it should be, since everyone knows what a dognitary is), it is a sniglet instead. This term was coined by comedian Rich Hall in the 1980's.

I would call that a form of "mash-up", as defined by oxforddictionaries.com:

noun, informal

a mixture or fusion of disparate elements

I'll hazard a guess with wordplay, or more specifically, pun.

I'll accept that "pun" usually requires a play to be made on two words that sound similar to each other. Whether the similarity is sufficient to warrant the definition pun in the example you give is open to challenge.