Native Finnish speaker here. Those are not postpositions but rather declension suffixes to the same stem word (koira, dog). It's still one word, and you discern it as such in speech. Though in speech, if the combination of suffixes you are using is particularly complex or unusual, you might need to make extra effort to speak particularly clearly to ensure intelligibility.

In English, you say "together with my dogs", in Finnish you could say "koirieni kanssa" (with-my-dogs together), or cram it all to "koirineni" (with-my-together-dogs), but you would pronounce the latter word clearly to ensure understanding as the -n- suffix in this context is uncommon in modern spoken Finnish.

Using a similar suffix method as Finnish uses to build declension, we could in English build from stem dog -> dogish (i.e. dog-like) -> dogishness (i.e. quality of being dog-like) -> dogishnessism ("ideology advocating the qualities of being dog-like"). It's clearly one word, but it might be hard to imagine a real everyday context to use it in.

Finnish is hard but not impossible. One great thing is the orthography is practically perfectly regular - something which English gets horribly wrong.