The word nur has the same sense as only, in fact auch nur ein Mensch has the same meaning as only human. As in that 80s song by the Human League.
In both languages it denotes that you're subject to foibles and fallibility.
Just like that famous Latin saying, supposedly by Seneca
errare humanum est
meaning, of course
To err is human.
Someone may say
Ich bin auch nur ein Mensch!
which, under pressure on a job and asked to work faster and better may mean I'm only one person, but also I'm not super-human.
When found out to have cheated, for an unfair advantage or in a relationship (as in the above-mentioned song), it may mean I succumbed to temptation.
But in this specific usage, it's a bit of a two-way street: on one hand the meaning as outlined above. Animals are fallible and make mistakes - so you may see that phrase used in some funny animal video.
But it's also a slight jab against humans that they too are just animals and not the supposed crown of creation (Krone der Schöpfung). Or at least I've always taken it to have that hidden connotation.
That's why there are also phrases like (courtesy @Crissov)
Tiere sind die besseren Menschen
Animals are the better humans/people
Frauen sind die besseren Männer
Women are the better men
The point being the somewhat ironic phrasing to link two disjunctive groups.