Think about it this way first: as one gets older, the risk of death increases. That's true whether the death is from a cardiovascular event or from another cause. That's true whether you are thinking about this in the context of a cause-specific model or a subdistribution model for competing events. There's no problem with a covariate like age at study entry affecting both events in the same direction, with either type of model. The question of interest might be which event does it affect more.

Interpretation gets tricky in subdistribution models, as the coefficients have a somewhat strange meaning to those who aren't familiar with them. Quoting from the paper you cite:

The exponentiated regression coefficient from a Fine-Gray subdistribution hazard model denotes the magnitude of the relative change in the subdistribution hazard function associated with a 1-unit change in the given covariate. Therefore, one is reporting the relative change in the instantaneous rate of the occurrence of the event in those subjects who are event-free or who have experienced a competing event. (Emphasis added.)

So the subdistribution hazards, at a fundamental level, aren't really for "competing events" in the way you might usually think about them. A subdistribution hazard represent the apparent hazard for a hypothetical population that includes both those who are event-free and those who have already experienced the other event!

I'd recommend also reading the treatment of competing risks for the R survival package. It turns out that interpretation of hazard ratios in subdistribution models can perhaps be more directly related to the ultimate risk of each type of event than those in cause-specific models, but they can lead to things like the summed probability of all event types exceeding 1.

Think about it this way first: as one gets older, the risk of death increases. That's true whether the death is from a cardiovascular event or from another cause. That's true whether you are thinking about this in the context of a cause-specific model or a subdistribution model for competing events. 

There's no problem with a covariate like age at study entry affecting both events in the same direction, with either type of model. It's often helpful to think of a 2-competing-risk model as a 3-state model. In that interpretation, the "competing risk" that's decreasing with age is the "risk" of staying in the event-free state. The question of interest might be which event type is more associated with age at study entry.

Interpretation gets tricky in subdistribution models, as the coefficients have a somewhat strange meaning to those who aren't familiar with them. Quoting from the paper you cite:

The exponentiated regression coefficient from a Fine-Gray subdistribution hazard model denotes the magnitude of the relative change in the subdistribution hazard function associated with a 1-unit change in the given covariate. Therefore, one is reporting the relative change in the instantaneous rate of the occurrence of the event in those subjects who are event-free or who have experienced a competing event. (Emphasis added.)

So the subdistribution hazards, at a fundamental level, aren't really for "competing events" in the way you might usually think about them. A subdistribution hazard represent the apparent hazard for a hypothetical population that includes both those who are event-free and those who have already experienced the other event!

I'd recommend also reading the treatment of competing risks for the R survival package. It turns out that interpretation of hazard ratios in subdistribution models can perhaps be more directly related to the ultimate risk of each type of event than those in cause-specific models, but they can lead to things like the summed probability of all event types exceeding 1.