It appears that -que was treated much like a word. Especially Ovidius does not treat it as an enclitic, but more as an independent word. This becomes evident in quotes, where -que is outside the quote but the word it is attached to is inside. Take a look at this question on a specific instance of this (for the version -c) and this list for a number of similar occurrences in Ovidius.
I am not sure how the Romans understood or defined a word. Word spacing as we know it — including our editions of ancient texts — was not an ancient phenomenon. Spelling conventions have evolved over time, also during the Roman era.
I don't know of any other ancient abbreviations where the "word" -que is given its own letter. But I didn't manage to find any other abbreviations containing -que either. In the absence of examples in either direction, it is hard to draw very strong conclusions. Modern abbreviations can be quite liberal, but arguing by modern analogies is dangerous.
One question worth asking is whether we really know what all the letters in SPQR stand for. The fact that the Temple of Saturn in the Forum Romanum has Senatus populusque Romanus spelled out in full (and so does the arch of Titus as luchonacho pointed out) is evidence in support of the standard reading, but it does not strictly rule out other options. However, I do find it most likely that the Q is -que and the standard reading is correct.
We can only guess why there is a Q, unless some ancient author discusses this very point. Perhaps it was to emphasize the "and". Perhaps -que was considered a separate word. Perhaps whoever came up with it didn't give it much thought and his decision stuck. Perhaps it was to differentiate it from Suomen Punainen Risti (The Finnish Red Cross). The last guess is probably bad, though.