Constants in the sense you mean are different to mathematical constants. They usually refer to a system in which two values are proportionate and so are of the units associated with the system.
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No, not all do. The proportionality constants that change the units will have units themselves.
All numbers are constants.
A constant is a value that never changes
A constant in an experiment is the object or value that does not change through out the experiment.
Numeric constants have the capacity to store numeric value.