I suppose it depends on how you define the two terms "foreign words" and "English morphemes." I would not call the English word "companion" a foreign word; to me that is just its etymological origin.
I think morphemes are generally considered a little more abstract and general than what you have in mind. In particular, I don't know of any requirement that they have to be apparent to linguistically naive native speakers. But in fact, Latin-derived morphemes such as the prefix com- are so common in modern English vocabulary that I think it's wrong to state that they never mean anything to English speakers. I also don't think there is any rule that all borrowed words are automatically mono-morphemic.
Regarding making new words from these morphemes (productivity): There are native English morphemes that are not productive, like the gain- prefix of gainsay. It's still considered a morpheme.
Taking companionship specifically
You agree with your professor that -ship is a distinct morpheme.
This gives us companion-ship.
Company vs. companion = a suffix -on or -ion?
The words company and companion seem to me to be related morphologically. If you agree, you can see that the "greatest common denominator" of these words is something like compan- or compani- (the latter works if we consider <i> and <y> to be purely graphical variants, and /i/ and /j/ to be allophonic or allomorphic variants that are conditioned by the presence or absence of a following vowel).
If companion and company are morphologically related, it seems to me that companion cannot be a single morpheme; it must have some other morpheme attached to the end. There are word pairs like relate/relat-ion-ship and un-ite/un-ion that show the putative -ion morpheme in a different context, so it seems reasonable to divide companion into compan-ion. (Note: "-tion" actually appears to be considered a morpheme by multiple linguists; “-tion in creation” is listed as a bound morpheme in these lecture notes for a Linguistics 101 course taught by Mark Liberman at the University of Pennsylvania. I suppose you could say that it carries meaning about the part of speech, since this suffix generally appears on nouns.) On the other hand, companion does behave a bit anomalously in some ways for a word with the suffix -(t)ion:
- normally this suffix takes the form -tion or -sion. The word union is actually anomalous in lacking a t/s.
- normally these words are abstract or represent the results of abstract processes; but companion refers to a concrete person.
- normally, the letters a, e, or o are given their "tense" or "long" pronunciations when followed by a single consonant (at the "underlying" level; that is, not counting single sounds represented by doubled letters like the "ss" in obsession) and the suffix -ion. For example, creation and abrasion. But companion is pronounced with a short/lax a.
If we look at the history, companion was not really formed with a suffix -ion. The hypothetical Latin ancestor compāniō is thought to have been composed of the prefix com-, the combining form pani- of panis "bread," and the augmentative suffix -ō(n-) (also seen in names such as Cicero). The Latin word passed into French, which is the immediate source of the English word.
Now, it doesn't make sense to me to analyze morphology on the basis of etymological knowledge that is not available to most native speakers. However, this information, plus the previously-mentioned anomalies if we interpret the suffix as -ion, does make me more inclined to analyse the word as compani-on.
Anyway, it doesn't affect the number of morphemes. This gives us compan-ion-ship or compani-on-ship.
Identifying the com- of companion as a prefix
As mkennedy and Adam say, com- exists as a prefix in many English words, usually meaning something like "together" or "with". Yes, most words with this prefix were taken from Latin whole rather than put together in English, but they are still English words now. So, we can separate this out as a distinct morpheme. Greg Lee points out another strong piece of evidence in his answer: the reduced quality of the first vowel. Apparently Chomsky and Halle's "The Sound Pattern of English" mentions the fact that morphologically simple words tend not to reduce the vowel of an initial closed syllable when the second syllable has stress: consider the pronunciations of "Montana," "spontaneous," "fondue," "bombard," "bombastic," "trombone," "conchoidal." Notice that there are reasons to identify "com-" is a separate morpheme even if we only look at other English words for comparison, not the original Latin words.
This gives us com-pan-ion-ship or com-pani-on-ship.
But is pan or pani really a morpheme in English?
I agree that it's questionable to identify pan(i) as a morpheme (especially one with the specific meaning "bread"), since I can't actually think of any other common English words that use this pan(i) "bread" morpheme (it's in pannier and panini, but neither of these seems fully naturalized to me, and it's in pantry, but not in a very recognizable form). A morpheme is supposed to be a meaningful unit, and it doesn't seem like native English speakers would associate any meaning with pan(i) on it's own.
This problem is not restricted to this one word, however. Two famous examples are the cran- and rasp- (or /ræz/ phonemically) in the words cranberry and raspberry. As far as I know, these are generally considered to be morphemes that don't have any meaning aside from their use in these particular words. But we can say that they are morphemes because berry is clearly a distinct morpheme.
With companionship, once we remove the clearly identifiable English morphemes com- and -ship, and the less clear but apparently present morpheme -ion or -on, we're left with what appears to be a morpheme pan, even if this type of analysis doesn't tell us what this morpheme means.
Edwin Ashworth's answer cites Language Through the Looking Glass: Exploring Language and Linguistics, by Marina Yaguello, translated by herself and Trevor A. Le V. Harris, which provides a different way of describing pan/pani that might be more useful or more appropriate. Yaguello speaks of "so-called paleo-morphemes, whose status and meaning are impenetrable for many speakers" (39). So maybe pan or pani is best classified as one of these "paleo-morphemes." It's unclear to me if Yaguello considers these a sub-class of morphemes: the next sentence suggests she does ("Such morphemes are all the more numerous in English, insofar as it is a hybrid language which has borrowed a great deal, in particular from the Classical and Romance languages") but the "turpitude" example cited in Edwin Ashworth's answer suggests to me that she doesn't. I'm not all that interested in terminological debates, but it is interesting to consider the limitations of the concept of the "morpheme."