Different solder composition affects the connection strength?
Good question! 

The short answer: if you are soldering bulk silver to copper, 63% Sn/ 37% Pb eutectic would probably be OK, but for silver plating, you might be able to make a successful join using tin/lead, if heat is quickly applied and removed, but might need to use a silver-bearing solder for thin silver plate or if the base metal under the silver is not easily soldered.

BTW, "cold soldered" is perhaps better expressed as low-temperature soldering, though both indium and gold can be joined at room temperature.

A longer explanation: reasons to use a specific solder composition.


To some extent, solder dissolves the surface of the metals to be united. This can be problematic with plated parts, e.g. gold plated over nickel, where the solder can completely remove the plating, making it difficult to adhere to the base metal. For that purpose, solders containing the plating metal are made, but they have other liabilities: expense and brittleness. Similarly, copper-containing solders are used for copper-plated electronic parts.
Not only alloys form between soldered parts, but also intermetallic compounds form, such as $\ce{Ag3Sn}$, $\ce{Cu6Sn5}$ and $\ce{AuSn4}$, some of which are brittle. Changing solder composition may not be practical to prevent some intermetallic compounds from weakening a joint, so barrier coatings and specific temperature cycling and storage may be used to retard their formation.
Pure tin solder, as well as zinc, has the property of forming whiskers, which not only short-circuit electronics, but was rumored to have caught Napoleon's troops with their pants down.


For these and other reasons, soldering can be considered as much a craft as a science. You may have to make some tests of the solder joints if they are critical, as in aerospace or medical applications.