Tate to Tate Ferry

Yes anyone can use the boat - it is part of the Thames Clipper/Uber boat service between Barking/Greenwich and Putney.
It costs £13.10 each way if you pay at the ticket machine at the pier, or £9.50 by contactless/oyster/on the Uber app, or £8.75 with a daily or weekly travelcard.

The fare falls outwith the daily/weekly TfL caps.

Thank you, Stuart, but are we talking about the same thing? I got the following off https://www.visitlondon.com/things-to-do/place/33114960-tate-boat-and-tate-to-tate-tours

Tate Boat & Tate to Tate Tours
It's easy to travel between the Tate Modern and Tate Britain galleries on the Tate Boat.
The Tate Boat travels between Bankside, Embankment, Millbank and St George Warf piers.
A Tate To Tate tour is available and includes a guided tour of the Tate Britain in the morning, a trip on the Tate Boat, and a guided tour of Tate Modern in the afternoon.
Prices and opening times
5-16 years old £2.50 to £3.75 per ticket
Family Ticket £36.00 per ticket
Concession Ticket £3.15 per ticket
Adult Ticket £5.60 to £7.50 per ticket

Yes, it's the same thing. It's a lot clearer and better explained on the Tate's own website, the giveaway on the Tate's website is the RB route numbers for the boat, and the correct fares.
Goodness knows how old that Visit Britain web page is, the picture is blurred but I don't even think it's one of the current generation of boats. It might even be a pre pandemic vintage page.

The link must be ancient because that is a huge increase in price.