Barclay is a Scottish surname used as a given name, meaning "birch tree meadow" from Old English. For a male dog it sits in the preppy, Old World register: the name of a dog with a monogrammed collar and strong opinions about where he sits. It belongs to a long tradition of British and Scottish surnames elevated to pet names.
Preppy Surname Dog Naming
Barclay fits the bracket of distinguished British surnames that work as dog names: Pemberton, Harlow, Beckett. Scottish Terriers are the obvious breed match given the Highland origin. Golden Retrievers with this name suit the country-club, outdoor-sporting owner type.
Banking Connection
Barclays Bank is one of Britain's oldest financial institutions, giving the name an inadvertent old-money association. Whether that adds charm or creates an unwanted corporate shadow depends entirely on the owner's sense of humor.
Counter-Reading: Sounds Like a Bank
The Barclays association will come up. Some people find it funny; others find it distracting. If you want the Scottish surname aesthetic without the financial institution shadow, Angus or Fraser cover similar territory more cleanly without requiring you to explain that your dog is not affiliated with any banking institution.