MySQL 8.0

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CREATE TABLE trans (
    account varchar(500),
    score integer,
    load_date date 
);

insert into trans values ('A1', 0,  '2021-01-01');
insert into trans values ('A1', 0,  '2021-01-02');
insert into trans values ('A1', 0,  '2021-01-03');
insert into trans values ('A1', 5,  '2021-01-04');
insert into trans values ('A1', 5,  '2021-01-05');
insert into trans values ('A1', 5,  '2021-01-06');
insert into trans values ('A1', 10, '2021-01-08');
insert into trans values ('A1', 10, '2021-01-09');
insert into trans values ('A2', 10, '2021-01-05');
insert into trans values ('A2', 10, '2021-01-06');
insert into trans values ('A2', 0,  '2021-01-07');
insert into trans values ('A2', 0,  '2021-01-08');
insert into trans values ('A2', 0,  '2021-01-09');
insert into trans values ('A2', 0,  '2021-01-10');

WITH cte AS (
    SELECT *, ROW_NUMBER() OVER (PARTITION BY account ORDER BY load_date) rn1,
              ROW_NUMBER() OVER (PARTITION BY account, score ORDER BY load_date) rn2
    FROM trans
),
cte2 AS (
    SELECT *, DENSE_RANK() OVER (PARTITION BY account ORDER BY rn1 - rn2) dr
    FROM cte
)

SELECT account, score, load_date, dr
FROM cte2
ORDER BY account, load_date;

account	score	load_date	dr
A1	0	2021-01-01	1
A1	0	2021-01-02	1
A1	0	2021-01-03	1
A1	5	2021-01-04	2
A1	5	2021-01-05	2
A1	5	2021-01-06	2
A1	10	2021-01-08	3
A1	10	2021-01-09	3
A2	10	2021-01-05	1
A2	10	2021-01-06	1
A2	0	2021-01-07	2
A2	0	2021-01-08	2
A2	0	2021-01-09	2
A2	0	2021-01-10	2