Short version: -eux might be the best candidate, as well as -ique, if you have to use an adjective construction instead of a paraphrase. Long version: read on. :)
Since knowing the technical name for a phenomenon always helps with research on it: this is a form of morphological derivation. French version of the article here.
A common derivational strategy is the one you've identified: suffixation.
This can be used to change the meaning or to change the part of speech. (For example, the English suffix -ful attaches nouns to form other nouns, even if it's not very productive now: handful, mouthful, etc. On the other hand, -ize can turn an adjective into a verb, e.g. regularize "make regular").
So now what remains is to find out which French suffixes turn nouns into adjectives, keeping in mind that no one affix can apply to every member of a class. After that, to identify which ones are still in active service and not just historical (many of them are remnants of Latin that are no longer productive). And perhaps finally to see which ones come out of native speakers' mouths the most often if you want to go for "naturalness". :)
Wiktionary French suffix category here.
One very undernourished subcategory: French adjective-forming suffixes.
This category includes only three: -able, -ais, -if.
A quick look at French words suffixed with -able indicates that this suffix attaches to verbs and usually results in the meaning: "which can have [verb] done to it". So not quite what you want.
A similar look at French words suffixed with -ais/e makes it clear that while this is for nouns, it only creates demonyms.
The best of these three for your question is definitely French words suffixed with -if/ive, where we see several nouns converted to adjectives, among them:
sportif effectif substitutif craintif fautif
Many more suggestions
That subcategory, however, is far from exhaustive.
For example, it doesn't include the highly productive -ique:
dogmatique canonique amnésique algorithmique ionique golfique
It also lacks the common -eux/se:
goutteux pierreux poussiéreux globuleux honteux cuivreux
And the possibly extinct -al/e:
régional architectural professoral parental dental
And of course the one English speakers borrowed from French and perhaps love more than the French themselves at this point: -esque.
titanesque livresque carnavalesque
It's also possible, though I don't think it's that common, to use past participle endings: -é and -u in particular. There are a bunch of words with this form that are nouns, especially -é (, though I suspect it's phonological coincidence with the actual suffix.
Looking through a few more of those suffixes from the higher-level Wikipedia category and checking the words formed from it would help build out this list.
Most natural choice
As for the second step, though, it's hard for me to tell which ones would come first to mind and attach most freely to a broad range of words the way -y does in casual English. It looks to me like -ique is a pretty clear winner in terms of number of derivatives, including obviously recent and off-the-cuff ones like jazzique. Edit: Frank suggests -eux as his suffixe préféré in the comments!
The problem, as you already noticed, is that many of these very common "derivations" don't feel like "actual words" and so won't show up in dictionaries or on lists like these. This is only really good for capturing the ones with a wide and long-lived acceptance.
If anyone has any suggestions, I'll happily add them to this answer. :)
Alternative ways to say the same thing
And then of course you have to consider other strategies than suffixes. For example, it might be easier to create a paraphrase than to come up with an instantly recognizable new adjective:
Ça sent bon / mauvais / doux / frais (adj.)
Ça sent chiennique ?
Ça sent le chien
While anyone might recognize what you likely meant by "chiennique", the other tactic is more naturally French (at least as I was taught it).
Edit: Another strategy in casual speech is simply to let the noun stand, as you might do in English:
There's a real "student" atmosphere.
Il y a une ambiance « étudiant ».
Here the quotation marks or guillemets, which help to show the bit of extra emphasis you might give it when speaking aloud, serve to set the word apart — as if to acknowledge and "apologize" for the fact that it's not adapted to the sentence structure. Notice that étudiant doesn't have to agree as feminine.
Edit: As others have now pointed out, a common paraphrase is de [noun] "noun-like". This is a great example of the translation technique of transposition: when the same idea is best represented by different grammatical structures in two languages. Here [adj] becomes [prep] [noun]. There are many such equivalences that translators find very important for rendering the target language naturally.
Appendix: morphological typology
errantlinguist suggests in the comments that French is a more analytic than synthetic language. Analytic languages, like Mandarin, have lots of individual particles (perhaps words) that each carry a bit of the meaning, while synthetic languages build or "bake" meaning into one base. I just encountered a word in the Hebrew Bible this morning that demonstrates syntheticism very well: w'hitzdaqtiu "and (w) I (ti) would (choice of vowels) do (hi) justice (tz-d-q) to him (u)". Particularly with that "choice of vowels" contribution, you can't pull this apart into separate words in a meaningful way. Clearly English is more analytic than Hebrew since I needed seven English words to translate one Hebrew word!
One thing that makes it hard to tell the exact status of French, English, and other languages is the issue noted above: how to tell how "productive" the affixes are. Latin was a fusional language, which is a type of synthetic in which a lot of grammar information is given by particular "shapes" the word takes. So you'd think Latin derivatives like French would also be fusional, but they differ in how much that holds true, and in what areas of the language.
A French verb's form gives information about the tense and the subject, and as Wikipedia points out, there are "45 different single-word forms of the same verb"! But that's still not as many as there could be. Why? Because to the ear, a bunch of forms sound exactly alike, so French adds a separate word to tell you exactly which subject it is: the je, tu, il, elle, on, ils, elles forms all sound alike for many verbs, so you have to say which one you mean, and you may as well throw in nous and vous once you've adopted the practice of having a separate subject. By contrast, Spanish, as you might know, "bakes" that information into the verb, which has no silent letters, so you almost never need to mention yo, tú, él, etc. to know who's doing the verb. So French verbs are more analytic than Spanish ones.
That was inflection — different forms of the same word. What about derivation, where you make new words? Well, all Latinate languages are packed to the brim with derivative affixes, but many of them are there just because the words are inherited from Latin, and the processes that derived the words aren't active anymore. As was pointed out in a comment above, canin is canis + -in. But what was canis in Latin is chien in French — and no one will accept chiennin. So canin, even though it can be broken down and analyzed, doesn't represent what French does now. On the other hand, the prefix re- "do again" is still alive and kicking, and it forms amusing words like rebonjour now and then.
So to sum up the relevant idea: there are a bunch of suffixes in French that clearly derive adjectives from nouns, as spelled out above. But telling which ones are still active and which ones are "dead" requires a look at (1) words formed recently, say the last 100 years, on these patterns, and (2) words formed on the fly that speakers would accept. If the commenters below are right, a French speaker today would be more ready to accept an analytic solution like de [noun] than a synthetic solution with a suffix for this sort of thing.
Of course, it's not 100% one way or the other (what is?), but it's something to think about. :)