There are multiple separate issues presented by this question.
Allocating Responsibility In The Lease
One is that the lease between Mike and Bob can attempt to allocate all responsibility for matters that could give rise to personal injuries at the rental property to Bob.
The lease could also require that Bob indemnify Mike for any claims that are brought against Mike related to the rental property. An indemnification clause basically makes Bob the insurer of Mike with regard to these claims.
Indemnification requirements are routine in leases.
Ultimately, it isn't clear that delegating responsibility for claim's on Mike's rental property to Bob could actually relieve Mike from legal liability to a third-party for these claims. Realistically, it probably couldn't because normally two people can't contract to limit the liability of one or all of them to a third-party who didn't agree to that. So, Mike would have residual liability for any claim that Bob couldn't pay.
So, it would make sense for there to be liability insurance in place to protect Mike directly as well, in addition to any insurance that Bob has in place.
Requiring Bob To Have Insurance In Force
The other issue is that the lease could require Bob to have insurance of particular kinds and amounts in place as a condition of the lease, to provide proof of insurance, and to set up the insurance so that Mike receives notice if it is canceled. Failure to do so would be a default under the lease.
Requirements in a lease that a tenant have certain kinds of insurance in force are routine in leases.
This said, while such indemnification requirements are common, it isn't obvious that Bob could obtain an insurance policy that would fully cover Bob's indemnification liability in this context as a matter of commercial reality, even though it would be legal to write such an insurance policy.
It would generally be easier for Mike to obtain liability insurance for the rental property on his own (which his mortgage, if he has one, almost certainly requires him to have in place) and to have the lease provide that Bob is required to pay for Mike's insurance policy on the rental property.
In certain kinds of leases, such as commercial "triple net" leases, it is routine to have the tenant pay the landlord's expenses related to the property rented. The indemnification requirement would also give Bob a strong incentive to keep his payments of the insurance policy protecting Mike current.
Alternatively, Mike could pay for his own insurance and increase the base rent by a like amount, which is economically equivalent to having Bob pay for Mike's insurance, and is almost economically equivalent to requiring Bob to have an insurance policy that covers Bob's indemnification liability to Mike.
If there was an insurance policy in place for Mike that is merely paid for by Bob, then the indemnification obligation of Bob would end up being limited to covering Mike's deductible for each claim under Mike's insurance policy, which would be a manageable uninsured risk for Bob to be responsible for paying.
If Mike were really worried about Bob's ability to pay the indemnification liability for the deductible under Mike's liability insurance policy, Mike could increase the amount of Bob's security deposit to make sure that Bob would have the ability to pay it out of the security deposit if from no other source.
Insurance Companies Almost Never Have Direct Liability For Third-Party Claims
It would almost never be the case that the liability for injuries on the property would be placed directly upon Bob's insurance company, or that Bob would be required to make a claim against his insurance company if he was sued. Liability attaches to people who own and do things. Insurance exists to help people who have liability or potential liability deal with that fact. Insurance companies are almost never (except in the highly specific case where an insured who has liability to someone for acts taken during life dies and leaves an insolvent and/or closed probate estate behind) a proper party to sue for liability for a covered liability.