The art & science of asking questions is the the source of all knowledge" - Thomas Berger

Do I have to consider weight, height and size (both my partner’s and mine) when I am shopping for a mattress?

Definitely.  Your weight, body shape and sleeping pattern should be well considered when you are shopping for a mattress.  If you are a side sleeper, it is recommended that you consider a slightly plusher mattress to allow for your side body contour.  If you are considered to be heavier person, a firmer mattress might be more appropriate for you.  More than often, for choosing a bed for you and your partner, there should be a compromise in comfort feel, so that both of your needs can be accommodated.