同じ先生から６日連続してレッスンを受けたって英語でなんて言うの？

You can use either of the sentences above to express yourself. It is always when you have the same teacher teach a lot of classes so that they are able to track you improvement and continue supporting so that you improve your English.

1. I took lessons from the same teacher six days in a row.
   This sentence is a basic and simple sentence, it means just what it says. For six days, you were
   taught by the same teacher.

The phrase 'consecutive days' means days that follow each other without a gap in between. For example, the six consecutive days may start from Monday and end on Saturday without a break in between. Any day may be the first day of lessons and the last day of lessons can also be any day, provided there are six days of lessons without a break in between. The adjective non-stop also means continuing without a break, without stopping.
So, you may say:
I took lessons from the same teacher for six consecutive days.
or
The same teacher taught me for six days non-stop.

When wanting to talk about something happening for a certain number of days, back to back, you can use the phrases: "consecutive days", "...days in a row", "...straight days".

If you want to explain that you took lessons from the same teacher for six days in a row, you can say:
"I have been taught by the same teacher for six days straight."
"I've been taking lessons from the same teacher for six days straight."
"I've had the same teacher for six days straight."

To say '6 days straight' or '6 days in a row' means the same thing - 6 days consecutively or after one another
So you would say 'I took lessons with the same teacher 6 days in a row/6 days straight' that explains it well.

You want to explain that you took lessons from the same teacher for six days in a row. Above are a couple of suggestions for that situation.