Yes and No.

Individuals had coats of arms and other heraldic devices (crests, mottos, etc). But generally they were used only by that single individual. His heir would inherit those arms, but wouldn't use them while his predecessor still lived. For example the coat of arms of the Earl of Northumberland was Or, a lion rampant azure, and only one Earl at a time would use it. The Earl of Northumberland was also the Baron Percy and the Baron Percy had a separate coat of arms, Azure, five fusils in fess or. Some Earls used the Lion, others used the Five Fusils. Most used a combination, where both the Lion and the Fusils would appear. Usually it would take the form of quartering the arms and putting the Lion in the upper left and lower right quarters and the Fusils in the upper right and lower left quarters.

However it was appropriate to create a new coat of arms based on one's parentage. A son of the Earl might take a modified version of his father's arms. These modifications are generally called a charge. And it would be a simple but clear distinction of the original coat. For example a chevron or stripe might be used. So while the Earl might be using a blue lion on gold, his sons might be using blue lions on gold with red stripes, one son using a vertical stripe, one a horizontal stripe, on a diagonal stripe, etc. All of these would be properly registered with the appropriate authority. So the family Percy riding to battle might all be carrying shields with a blue lion on them. But each shield should distinct and recognizable. A well trained herald should be able to distinguish each lion of the north from The Lion of the North.

Sons might also take and combine their father's arms and combine them with their maternal grandfather's particularly if they had a claim to some sort of to make along that line. This could result in some coats getting very complex. For example here is the current coat of arms of the current Duke of Northumberland. It contains not only the old coats of arms of the Earl of Northumberland and Baron Percy. It also contains the coats of Branant, Lovaine, Lucy, Drummond, and Drummond augmented. Several of these aren't the original coats either, but contain charges. For example the center bottom "grandquarter" is a traditional Drummond coat, but the red diagonal stripe is a charge to it.

While the /u/QuickSpore provided an overview of English heraldic traditions, it's worth to note that in other countries the customs could be rather different.

For example, Polish nobility did, in fact, have 'family sigils' and 'mottos' to a certain degree, though both of these elements were different from their Game of Thrones equivalent.

Most notably, coats of arms and "proclamations" (Polish zawołania) were shared on the Clan (ród) level rather than being different for every individual family.

Secondly, zawołanie differed from mottos in that they were originally used as a battlecries, so they were kept short and to the point.

Most of them sounded rather banal as well, usually referring to the local area/strongold/towns (Game of Thrones equivalent would be Arya yelling 'Winterfell!" in the heat of a battle), though few were a bit more creative (like the one that could be translated as Tear off their hoods!)