Create line chart without aggregation

You need to add a dimension that splits those individual weights into different ones. For example, datetimes when the weights were taken which will serve as a group of each individual weight. If you don't have a datetime column, you can add an index column then rank the weight value for each date/person/index.

Notice that there are two rank valeus for 2/1 in the above screenshot.

You can add another calculated of invisible characters to be added to the visual so the weights can be split into different categories and not just by date.

Notice in the screenshot below there is a wider gap for 2/1 as it consists of two rank values

Please note that another column to X-Axis changes its type from continuous to categorical (continuous can be used only on date, datetime and number data types only). Please see the attached sample pbix.

Addendum:

If each record regardless if the weights are the same must be shown separately, ranking must be based on the index column

RANKX (
    FILTER (
        'Table',
        'Table'[Date] = EARLIER ( 'Table'[Date] )
            && 'Table'[Person] = EARLIER ( 'Table'[Person] )
    ),
    [Index]
)