No. It's not a difference in degree or feeling, it's a logically different verb. Notice that there are two kinds of sentences in English that use "know."
(1) "I know that it's raining", "I don't know whether he's alive", "Did you know that zebras have stripes?"
(2) "I know John", "I know Paris very well", "I don't really know those guys"
In (1) the word "know" is followed by a fact or piece of information. In French these cases are translated with "savoir." "Je sais," etc.
In (2) the word "know" is followed by a noun phrase, an object you know. We might also say "I'm familiar with" or "I'm acquainted with," but not because of how personal it is. Rather, this verb expresses a relation to an object rather than to a fact. In French this is "connaitre."
(German has the same distinction with wissen and kennen. English also used to have it, but it's long gone now.) For clarity, in your head, you might try associating "savoir" with "I realize that . . ." or "I am aware that . . ."; and translate "connaitre" with "I have encountered . . ." or "I have heard of . . ."
I put "You understand animals" as it made more sense than knowing them as in "This is Harry the horse and Betty the cow", but was marked wrong. Should "understand" have been accepted?
Does the use of connaitre in this case, as opposed to savoir, imply that the animals are known on a more personal level?
savoir is more related to knowledge and ability, connaitre to understanding or personally knowing as far as I can tell.
I think it is possible that this sentence could be implied to mean that the animals are known on a personal level. I also think that this sentence is being asked while it is being stated because the speaker could be wondering if whom he is talking to knows the animals. To answer your question, I think that the sentence could be on a personal level but, doesn't have to be.
Connaitre means to be familiar with, je connais moutarde de dijon Savoir only really applies when talking about knowing facts. Je sais il est grand