You can’t be signed in using multiple email addresses at the same time in one browser. Your best bet is to have each account open in a different browser or use your browser’s private/incognito window feature to sign in to an additional account.

If you don’t want to merge two Harvest accounts into one Harvest ID because you own separate companies with different email addresses, note that invoices sent to your clients don’t need to appear to come from the email address you use to sign in. You can set a different reply-to address in Invoices > Configure > Company information; you could then merge your email addresses to take advantage of the benefits of Harvest ID while continuing to make your invoices appear to come from two different email addresses and companies.