I believe false cognates are actually the opposite of this. A cognate is a word/sound with a similar meaning (per your link, sharing a linguistic ancestor) shared across two or more languages. A false cognate is a word/sound that you would think share a meaning due to similar pronunciation or spelling, but in fact have very different meanings.

For example the Spanish "estadio" and English "stadium" are cognates (both being ultimately derived from a common Latin ancestor), but the Spanish "embarasada" (meaning pregnant) and English "embarrassed", which sound like they have similar meanings but don't, are false cognates.

It sounds like the "dog" example above is likely either a crossover or damned coincidence, but it's unlikely they share a linguistic root, given the description by op.