Did the Romans literally think the planet was an embodiment of the god? Or did they think the planet 'belonged' to the god, or otherwise symbolized it? (Obviously the same question applies to any planet/star, not just Jupiter)
They were named after the characteristics of their gods. Some examples:
There are some interesting factors in the naming of the objects. Jupiter's satellites have naming conventions which consider their orbit - for example ending in e if they are in retrograde orbit.
Source: International Astronomical Union: Working Group for Planetary System Nomenclature
"The largest and most massive of the planets was named Zeus by the Greeks and Jupiter by the Romans; he was the most important deity in both pantheons." Found in source cited above, under "Jovian System".
Edit:
As pointed out by many, this explanation already got critisism in the thread I linked at the top. There, /u/rosemary85 pointed out that [there are indications pointing to derivation of the Greco-Roman names from Babylonian astronomy] (https://www.reddit.com/r/AskHistorians/comments/1iaxt2/what_was_the_relationship_between_ancient_greek/cb2p3yd) and the characteristics didn't have anything to do with the naming.
Now, these [planet names] aren't names of Babylonian gods; but the tablet does refer to Sihtu (Mercury) as "the star of Ninurta", and Neberu (Jupiter) is "the star of Marduk" -- and these are known Babylonian gods. So the planets had names and associations with particular gods.
The Greeks didn't think that planets were embodiments of their gods. They thought planets were wandering stars, as the word planet is derived from ἀστήρ πλανήτης (astēr planētēs), which means wandering star.
In astrological texts written in Greek during the Roman Empire the names of the planets were often given in the genitive case, which is the possessive case. So, they didn't just say "Zeus" or "Jupiter" typically, but they say "the star of Zeus" or "the star of Jupiter". This implies that many of them thought that the planet belonged to the god, rather than literally being the god itself.
For an example of this see the first chapter of book 1 of the Anthology of the 2nd century astrologer Vettius Valens:
https://books.google.com/books?id=OQ_gAAAAMAAJ&pg=PA1#v=onepage&q&f=false
Follow-up question: how did they know they were looking at planets without modern optics?
I have a follow up question. Was it because the Greeks and Romans were unable to figure out why the planets moved the way they did that they decided they must represent gods?