Some did, although not all; and individuals might adopt their own personal motto on top of the family motto. Generally, until the trend in favor of meticulous regulation of heraldry started in the late fifteenth century, their use was pretty varied and inconsistent.

In my own area of interest, pre-modern Italy, many urban governments incorporated on their heraldry the phrase "LIBERTAS," meaning "Liberty," starting from the eleventh century. The intent was to signal their emphasis on consensus-based governance and opposition to higher authority. Other Italian cities incorporated into their heraldry religious phrases that could be also interpreted as a kind of "motto." Venice, for example, adopted "PAX TIBI MARCE EVANGELISTA MEUS," words purportedly heard by St. Mark the evangelist as he travelled through the lagoon and witnessed a divine vision.

Of course, this being the "middle ages," where would we be without the great landed families? ("In more or less the same place," says the institutional theorist, scratching his head, but that's beside the point) Reflecting the relatively inconsistent use of mottos, many families only adopted a motto relatively late; the Medici, for example, only adopted "Festina lente" ("Hurry slowly") halfway through the 16th century. Cosimo de' Medici, Grand Duke of Tuscany, originally made it the motto of his fleet, however subsequent Medici seem to have been rather proud of the ironic-sounding advice and it appears on a number of public buildings in Florence and Tuscany.

The Visconti of Milan offer a more colorful example. Originally, their motto was "Vipereos Mores Non Violabo," roughly translatable as, "Don't Violate the Viper's Laws" (perhaps more loosely translatable as, "Do not Disobey the Viper). With a coat of arms depicting a serpent eating a man, the message they're sending out seems clear. However, the Visconti were fluid about their use of a motto; Gian Galeazzo Visconti, while heir to the Duchy of Milan, adopted his own motto after his wedding to Isabelle de Valois, a French princess. Gian Galeazzo's words, "A bon droit," are french phrase meaning, "With good right," but more accurately translated as meaning, "With good reason." After Gian Galeazzo use of a motto seems to fall out of fashion until the Visconti are succeeded by the Sforza in 1466, a female-line branch of the family. Galeazzo Maria Sforza had the phrase, "Io non dimentico," Italian for "I don't forget," inscribed in stonework commissioned during his reign; a somewhat confusing motto, and perhaps more apt for his brother and successor, who instead is sometimes attributed the words, "Per Italia nettar d'ogni bruttura," meaning (more or less), "To clean Italy of all ugliness" (although this might be a later fabrication by artists who had benefitted from his patronage).

There are many other colorful examples: the 14th-century mercenary captain Guarnieri d'Urslingen adopted the motto, "Nemico di Dio, della pietà e della misericordia," meaning "Enemy of god, pity and mercy." However, there are just as many mottos that are rather bland; the House of Savoy's motto was (and still is) "FERT," which is not a word in any known language and no one knows what it's supposed to mean.

I found a previous thread on this with lots of info from /u/Quickspore and others.

To summarise, English heraldry tended to be for individuals, or to pass down with modification from father to son (for example) but other European countries did have mottos for extended families or clans. Go to the Original Thread for a better explanation.