When Dutch gains a target lock he can allow another ship to get a target lock on the same ship ignoring range. If the second ship that gains the target lock has R3 astromech, which allows you to gain two target locks on different ships, can the second ship get another target lock on a different ships and ignore range restrictions?

no.

r3 only allows a ship to gain 2 target locks when it preforms a target lock action. Dutch just allows them to acquire a lock.

However if a ship with r3 already had a lock it could acquire a second lock without losing the first

I would say no.

Dutch allows the second ship to acquire a lock

"After you perform the action, you may choose 1 friendly ship at range 1-3. That ship may acquire a lock on the object you locked, ignoring range restrictions."

R3 requires you to perform a lock action

"After you perform a Target Lock action, you may acquire a lock."

The two answers above me are correct.

R3 Triggers after a Lock Action . Dutch's special ability is "acquire a lock." R3'll still allow a pilot helped by Dutch to maintain a second Lock, if you'd already had a lock from before, but Dutch with R3 doesn't hand out two locks, and a ship helped by Dutch doesn't acquire two new Locks.