Because fname can be both a string or a file-like, this line reads a little strangely. Would you mind adding a comment? Something along the lines of:

if we've been given a file-like object, and it has a name defined, get the extension from the name.

Sorry to quibble, but I think that in the context of the few lines above, little or no comment is needed. At most, "file-like object: check for name with extension" or something like that.
This changeset leaves in place the likelihood of somewhat obscure error messages if the file name lacks an extension.
One could use something like this:

try: # file name
    basename, ext = os.path.splitext(fname)
except AttributeError:
    try:  # file-like object
        basename, ext = os.path.splitext(fname.name)
    except AttributeError:
        pass
try:
    ext = ext.lower()[1:]
except IndexError, NameError:
    ext = "png" # No extension, or no name provided: try png.

Too ugly?

Too ugly?

There is a lot to hold on to in your head. And even then passing a fname='broken.' would result in no extension being identified.

Apart from the (admittedly necessary) use of is_string_like, I have no major problem with this code block. To address the IndexError you are referring to, I would probably factor out the indexing and adding a line before if ext not in handlers.keys(): to the tone of:

if ext.startswith('.'):
    ext = ext[1:]

And depending on if we actually care about users who might pass format='', I would probably drop the else: ext='png' out of the if block and do:

if not ext:
    ext = 'png'

Later on.

Of course, this is a backport, so I'm not really advocating a structural change for this PR. I do think, however, a test would be beneficial.