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This is the default (and the only) behavior. It is not explicitly documented, but is implied by systemd's operation logic.

systemd.timer(5) reads:

For each timer file, a matching unit file must exist, describing the unit to activate when the timer elapses.

systemd(1), in turn, describes the concept of unit states and transitions between them:

Units may be "active" (meaning started, bound, plugged in, ..., depending on the unit type, see below), or "inactive" (meaning stopped, unbound, unplugged, ...), as well as in the process of being activated or deactivated, i.e. between the two states (these states are called "activating", "deactivating").

This means that the triggering of a timer leads to "activation" of the matching unit, i. e. its transition to the "active" state.

If the matching unit is already "active" at the time of "activation" (for a service unit, this means "the main process is still running", unless the service unit has Type=oneshot and RemainAfterExit=true), it should be obvious that no action will be taken.

The solution was quite simple: one has to replace

[Service]
Type=oneshot

by

[Service]
Type=simple

This doc states:

Type=simple (default): systemd considers the service to be started up immediately. The process must not fork. Do not use this type if other services need to be ordered on this service, unless it is socket activated.

...

Type=oneshot: this is useful for scripts that do a single job and then exit. You may want to set RemainAfterExit=yes as well so that systemd still considers the service as active after the process has exited.