Vocabulary – ice hockey etc – does one ' "score" two assists'?

In soccer, I think the usual verb is 'have'.

'V an assist' is relatively new. But there are almost 3½ million Google hits for "had three assists" and over 5 million for "had two assists", many obviously relevant, for instance.

Here are, for comparison, various other verbs which are logical candidates found in raw Google data for "Ved two assists":

• had two assists > 5 million
• scored two assists 640 000
• totaled/totalled two assists 15 000
• made two assists 1 000
• achieved two assists 1 000
• amassed two assists 1 000

and the multi-word verb

• came up with two assists 133 000

There will be false positives, of course, and raw Google data is not too reliable, but there's a clear preference, with a clear second and third. Then again, some sentences might work better with a verb other than have. This colligation (preferred V-N pairing) example is, as said, new, and just settling down.

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Note that your first example

• "He scored three points in the game, two goals and one assist"

does not have a compound object (as in "He scored two tries and a conversion") but an enlarging/explaining adjunct. A colon could (and often would) replace the comma. This means that neither 'goals' nor 'assist' has to be an accepted direct object of 'scored'. "I made friends with two girls in France: one Irish, the other Pakistani." However, in ice hockey, this appears to be an appositive as both goals and assists (up to 2 for the latter) score 'points' for players. Lambie's answer gives this interpretation; it's getting into jargon, but is probably on-topic as many follow ice hockey.

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Also note that "He scored two assists" in soccer has a diferent subsense for 'score' than "He scored two goals". If A scored (made) two assists and B scored a goal from each of them, a total of 2 goals only resulted. Two goals were scored (in the normal sense).

The context is ice hockey:

1. And one sees: to get an assist or more formally, to be awarded an assist, to be credited with an assist.

Assist

2. USA Hockey

Up to a maximum of two assists shall be awarded to those player(s) who had possession of the puck immediately prior to the goal being scored.

National Hockey League

You score a point when you shoot the puck through the goal posts, and whoever helped you is given or awarded an assist.

• He scored two goals and got or had two assists in the game.

score is only for goals. Not for assists.

However, you can say: He was awarded two goals and two assists.

(When a goal is scored, there can be up to two players who get assists. Assists are recorded for a player.)

I just heard on TV re the Bruins: He made two goals and four assists.

Yes. In hockey, an assist is scored as one point in the assisting player’s record. This is from the National Hockey League:

When a player scores a goal, an “assist” shall be credited to the player or players taking part in the play immediately preceding the goal, but no more than two assists can be given on any goal. Each “assist” shall count one point in the player’s record.
_{Source: NHL—Rule 57 Goals and Assists}

And here are some usage examples:

McAuliffe led the way with four points, scoring one goal and three assists in Harvard’s furious second period strike.
_{Source: The Crimson—No. 10 Princeton Shocks W. Hockey (2004)}

In his NHL career that spanned 18 years, Carpenter played 1,178 games, scoring 320 goals and 408 assists for 728 points.
_{Source: Wikipedia—Bobby Carpenter (ice hockey) (2006)}

In 1942, at the age of 21, Richard scored an assist in his first game with the Montreal Canadians and then a goal and an assist in his second.
_{Source: The Pink Puck—This Day in Hockey History... (2019)}

Ken Willis is still high scorer, having ten goals and two assists for 22 points. . . . A composite score including goals, assists, points, and minutes in penalties follows . . .
_{Source: Princeton Alumni Weekly, Volume 35 (1935)}

Note that assist scores are not part of the game score, which counts only goals.