Do pentesting tools leave behind any information in order to identify the attacker?
Yes, and no.
Yes, they do leave behind log entries, TCP and UDP connections, login attempts, a lot of forensic data. Depending on the tool, it's possible to determine which tool is being used and what the attacker intends to do.
No, it's usually not possible to identify the attacker. If the attacker uses its own IP address, you can have an IP address. Linking that IP to the person behind it takes a lot of work. If there was little or no damage, it's almost impossible to do anything about an attack.
If the attacker does not used its own IP address, but logged in at a public WIFI network, used Tor or another open proxy, you will not be able to identify the attacker. It's possible, but not for you or me. International justice systems have to agree on the search, different law enforcement forces from different countries have to work together, and takes a lot of time (months, years) to identify the person. And if the country of residency of the attacker is on a jurisdiction that does not cooperate with yours, you can do nothing.