Can babies get insomnia?

Hard to define. I am convinced babies don't understand or appreciate sleep until after 1yr. Well babies drift between sleep/wake states based on their feed/activity/sleep cycle.They fade into sleep like adults in a boring lecture, so stimulating a tired baby by rocking may prevent them from settling. Overly tired infants want you to take away their tireness. Your best bet is to let them learn a good pattern.
Yes, but... Insomnia is hard to define - usually relates to difficulty falling asleep or staying asleep. I would want to know how old the baby is - until 3-4 months, many babies have erratic sleep patterns. After that, most babies settle into more predictable sleep patterns. If your baby can't self-soothe and settle to sleep at least 10 hours at night by 4-6 months, that could be called "insomnia".