Unique Key Violation in SQL Server - Is it safe to assume Error 2627?

Within an approximation, yes.

If you search the MS error and events site for SQL Server, error 2627, you should hopefully reach this page¹, which indicates that the message will always concern a duplicate key violation (note which parts are parameterized, and which not):

Violation of %ls constraint '%.*ls'. Cannot insert duplicate key in object '%.*ls'.

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¹As @2020-06-18, Database engine errors and events would be the correct page to go to

2627 is unique constraint (includes primary key), 2601 is unique index

SELECT * FROM sys.messages
WHERE text like '%duplicate%' and text like '%key%' and language_id = 1033

Here is a handy extension method I wrote to find these:

    public static bool IsUniqueKeyViolation(this SqlException ex)
    {
        return ex.Errors.Cast<SqlError>().Any(e => e.Class == 14 && (e.Number == 2601 || e.Number == 2627 ));
    }