What you have is an individual’s coat of arms. The whole point of a coat of arms was to identify the person displaying it when he was concealed by armour; that doesn’t work if more than one person is entitled to carry the same arms.
Suppose a coat of arms is granted to Joe; Joe is the only person entitled to use those arms. Joe’s children, if they want to use arms, will usually use what are called “differenced” arms. Strictly speaking, they should apply to the appropriate heraldic authority for a grant of arms of their own, and normally they will be given their father’s arms, “differenced” by the addition of extra elements (“marks of cadence”), though there is no fundamental reason why they wouldn’t get radically or completely different arms. In practice, though, it wasn’t uncommon for people simply to adopt their own “differenced” arms by adding the conventional marks of cadence.
When Joe dies, his arms pass to his heir – his eldest son, if he has one – and he can use them without a grant from the heraldic authorities. All Joe’s other relatives need their own grants, and they will not be granted Joe’s arms, but something different.
In theory, arms which have been “differenced” with marks of cadence, and which have been granted by heraldic authorities, are also heritable, and the process can be repeated from generation to generation. But this means marks of cadence superimposed on marks of cadence, and makes for very cluttered arms. In practice, if a few generations of younger sons were important enough in their own right to bother using arms, they would apply for their own arms, not their father’s or grandfather’s arms differenced by marks of cadence.
In 1390, William Grosvenor was granted arms consisting of a sheaf of white on a blue field. Those are now the arms of his heir, Gerald Grosvenor, the current Duke of Westminster. He is the only person entitled to use those arms. All of William Grosvenor’s other descendants – and there are many – either use slightly different arms, completely different arms or – in most cases – no arms at all.
The arms of the head of a (significant) family tend to be identified as “the family arms” but, strictly speaking, only one person is entitled to use them at any time.
Can arms be identified? Well, that’s the whole idea so, generally, yes. But of course the more significant the person entitled to bear the arms, the easier they are to identify. The heraldic authority which granted the arms should be able to identify them but, depending on the country concerned, it may no longer exist. If you only have partial arms – no motto, possibly other elements also missing (supporters? helm?) and perhaps no indication of colouring – definitive identification may not be possible. On the other hand, if you know the provenance of the article on which you found the arms, that could narrow down the possibilities.