Learn English – Name of small company gifts (as marketing tool)

I cannot speak to English elsewhere, but in the US there would be no need to use size at all—‘small business’ is the term.

If you are concerned that bare small suggests inferiority, quite the opposite is true. ‘Small businesses’ are generally recognized—or at least recognize themselves!—as the engine which drives the American economy, and there is a large Federal agency, the Small Business Administration, dedicated to their support. They have indeed something of a mystique: ‘small business owners’ regard themselves as a distinct class, superior in efficiency, energy and entrepreneurship to the bureaucratic managers of large corporations.

Of course the US notion of ‘small business’ may be substantially larger than that which obtains in your country. The SBA standard varies from industry to industry, but by and large it is less than 500 employees in manufacturing and less than $7M annual revenue in non-manufacturing industries.

I assume you want to designate the relationship between them in order to, for example, describe it (as good, bad, cooperative, competitive, etc.).

The relationship between company x and company y is correct. We normally do not use the other way.

Between is cited here in the example sentence provided for the relevant sense of relationship:

3. an emotional or other connection between people: the relationship between teachers and students.

Some English learners may feel that the relationship of x and y is correct because we sometimes use [something] of [something] similarly to 's. For example the entrance of the building, a friend of my grandmother. But we do not generally use this form for a possession of the type X and Y's relationship.