This comes up a lot these days, note that the "Send As" changes in EXO, also do not get confused with the differences, shown below:
Send on Behalf as : This is usually when people use Outlook to add a "from" field, as you are sending on behalf of your mailbox.
Send As : This is when you connect to the mailbox directly and "sendas" that mailbox for example if Junior Bear and doing a SendAs from Grizzly .Bear - usually you would visit the "another mailbox" URL like this > https://outlook.office.com/Grizzly.Bear@bears.local
These are the commands you need:
Send on Behalf
This will grant Junior.Bear permissions to send on behalf of Grizzly.Bear
Set-Mailbox Grizzly.Bear -GrantSendOnBehalfTo Junior.Bear
Send As
This will grant Junior.Bear permissions to send as Grizzly.Bear
Local Exchange : Add-ADPermission Grizzly.Bear -ExtendedRights Send-As -user Junior.Bear
EXO : Add-RecipientPermission -Identity "Grizzy.Bear" -Trustee "Junior.Bear" -AccessRights SendAs
Full Mailbox Access
This will grant Junior.Bear full access to Grizzly.Bear's mailbox.
Add-MailboxPermission -Identity Grizzly.Bear -User Junior.Bear -AccessType FullAccess -AutoMapping $False