As written in Kremer, 1997 and Štekauer, 1995 words like ‘cat-like’ or ‘congressman’ etc. have what they call suffixoids or semiaffixes. IMO, it’s kind of a dull idea, because they look like straightforward words consisting with two stems joined by the null interfix, am I right?
Note that there is always a grey zone between suffixes and compound words. Arguably, all suffixes were independent words historically, but lost their independent meaning (in German verblassen "to bleach out"). So we have at any time some things that are still recognisable as words, but half-way on becoming a true suffix. This things are called suffixoids.
Suffixoids and prefixoids are words typically referring to some middle stage of grammaticalisation, i.e. when we do not consider the element a stand-alone word but it is not comparable yet to regular derivational morphemes.
This seems like a pretty apt analysis of both -like and -man in the examples above.
For -like, this is something that has actually already happened in Germanic languages in the past and it is the source of the adverbial suffix -ly/-lich/-lik etc. You can argue that -like serves as a derivational suffix because you can convert it further, e.g. to abstract noun (catlikeness - I supposed this would not work as a derivation from a compound word... cat+likeness).
For -man, you can actually even see that this form has been reduced phonetically - it lost accent and the mid-vowel changed to schwa. This is arguably being deconstructed lately by gender-conscious language as many people replace this element by -woman or -person.