Are strawberries to be planted vertical? On a trellis or cage?
No. Although some people plant them in a hanging pot.
I saw someone with a bunch of pots stacked up and they planted the strawberries in the top pot in year one and by year 3 they had all of the lower pots (4 more I think) filled with strawberries by directing the runners from the above pot into the lower pots. It looked pretty cool and produced a ton of berries.
I've never seen any that climb upwards like a vine which is what I think you are asking.