We should be careful to avoid cluttering the interface by adding too much information.

Consider the case where a single issue is updated several times in sequence, either via distinct manual changes, or with a single event that generates multiple history entries (e.g. resolution: status change, assignment, note added). Displaying many times the project name could make the timeline less readable.

At the very least, the project name should only be shown when necessary (i.e. when it is different from the currently selected project).

At the very least, the project name should only be shown when necessary

right

I was thinking the same as you, how to do it to not clutter
A good place may be to add it to the text that pops up when hovering over the id link.