The motto of the Roman Republic was, of course, Senatus Populusque Romanus, or SPQR. However, Romanus is a masculine, singular adjective. What confuses me is that it is referencing Senatus Populusque. While each might individually be singular, together as a combined nominative, they are considered plural. So, shouldn't the adjective be the Romanus but plural, in this case Romani?
Take a look at this older question for gender and number of an adjective referring to several nouns. There are two basic cases. The adjective can be attributive or predicative. If you want to say "The senate and people of Rome did this and that", the adjective "of Rome" is attributive. If you say "The senate and people are Roman", then the adjective is predicative. Predicative adjectives tend to come with esse.
In the attributive case the adjective agrees in number and gender with the nearest noun. In the predicative case the adjective is plural and the gender depends on the genders of the nouns; see the linked question and its accepted answer.
In SPQR the adjective is used attributively. It is intended to be a noun phrase, not a statement that the senate and people are Roman. Therefore it should indeed be Romanus, not Romani.
As to why the phrase Senatus populusque Romanus is abbreviated SPQR and how we know what the letters stand for, see this question.