Without looking at your mortgage paperwork, it's hard to know. Were I to guess, though, there are often expenses that appear both as prepaids and as escrows paid at closing in your closing documents in addition to being part of the regular escrow account. In other words, you're making an initial payment prior to closing, then make a payment as part of the closing, and then making regular payments from the escrow account. For example, on the CFPB's sample closing disclosure, you see on page 2
that 12 months of homeowner's insurance was prepaid and 2 month's of homeowner's insurance premiums were deposited in the escrow account at the time of closing. If this mortgage closed today, the insurance company would have 12 months of premiums initially (so the house would be covered from 3/2020 until 3/2021). The escrow account would then accumulate the initial two months worth of premiums and would have enough money to pay the next year's premium in 3/2021 but you might not see a payment from the escrow account for a year.
Even if nothing was prepaid, though, if the insurance company is telling you that you're covered, you should be fine. It's very common for banks holding escrow accounts to make transfers once a month (or a handful of times a month) to a particular recipient and to alert the recipient to upcoming payments outside of that. If the escrow company sends the electronic equivalent of
"We've got 201.66 in John Doe's escrow account for Homeowner's Insurance at $100.83/ month based on the closing of loan 123456789. We'll be transferring that payment along with 52171 other payments on 3/31."
and the insurance company knows that the money has safely been paid. One benefit of having a third party escrow the money is that the payee knows that the money has already irretrievably left the payee's account so the risk of non-payment is effectively zero. You might not see the payment actually leave the escrow account until 3/31 but the recipient knew that it was already paid as of closing.