A regional navy needs influence within its region

Does that make every European navy except Britain's and France's a small navy?

No; a regional navy can exert power across its part of a region without having global reach.

European fleets outside Britain and France need not all fall into the small-navy category. Regional navies can project force across a wider area while lacking sustained worldwide reach. They do not have to dominate every navy in the entire region; the relevant capability is influence over their own part of it.