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Definitely not a requirement but they tend to be relationship-oriented so it helps.
Banks will not require an existing relationship for approval for a credit card. They may require an existing relationship for some of their higher end cards and may have a existing relationship be more lenient for you on other product. Some banks are however region locked which might feel similar. Most banks do not care about having an existing relationship as they prefer to expand by the issued product.
The only financial institutions that require a relationship are credit unions. This is because they wish to only serve their members.