One way to break an outer loop from an inner loop without adding an extra variable to keep track of is to use else: continue followed by a break:
while True:
    user = input("Enter: ")
    if "/" not in user:
        if "," in user:
            date = user.replace(",", "")
        date = date.split(" ")
        date = list(date)
        for i in range(len(date)):
            if date[i] in month:
                mnth = month[f"{date[i]}"]
                if int(date[1]) < 10:
                    date[1] = "0" + date[1]
                print(f"{date[2]}-{mnth}-{date[1]}")
                break
        else:
            continue  # continue if the for loop wasn't broken
        break         # break the while if we didn't continue
As explained in the comments, the trick is that the else of the for executes if there was no break -- we can therefore continue the while if and only if there was no break inside the for, and explicitly break the while otherwise.
Not directly related to that, I'd suggest battling some of that indentation by using continue elsewhere in the loop, and getting rid of all the unnecessary string manipulation (note the use of {date[1]:0>2} to zero-pad to two digits without needing the int math -- neat, eh?):
while True:
    user = input("Enter: ")
    if "/" in user:
        continue
    date = user.replace(",", "").split(" ")
    for mm in date:
        if mm not in month:
            continue
        print(f"{date[2]}-{mm}-{date[1]:0>2}")
        break
    else:
        continue
    break
and since your code seems to assume that date[2] is always the year and date[1] is always the day of the month, I'm going to go out on a limb and guess that you can also safely assume the month is always date[0] if it's anything at all, which means you can skip that whole inner loop and just do:
while True:
    user = input("Enter: ")
    if "/" in user:
        continue
    mm, dd, yyyy = user.replace(",", "").split(" ")[:3]
    if mm not in month:
        continue
    print(f"{yyyy}-{mm}-{dd:0>2}")
    break