In some jurisdictions this would be theft of services, or some other variant of theft.
Theft of services often includes accessing utility services without payment, particularly by evading metering; "turnstile jumping" to avoid payment for transportation services; watching a movie or performance (such as in a theater, concert, or circus) by evading the required payment.
In some jurisdictions there is no separate crime of "theft of services", it is simply one fact pattern for a crime of theft or larceny.
In ohio, for example, Section 2913.02(A) of the Ohio Revised Code provides that:
No person, with purpose to deprive the owner of property or services, shall knowingly obtain or exert control over either the property or services in any of the following ways: ...
(2) Beyond the scope of the express or implied consent of the owner or person authorized to give consent;
which would seem to apply to the situation described in the question. However Section 2913.01(E) provides that:
(E) "Services" include labor, personal services, professional services, rental services, public utility services including wireless service as defined in division (F)(1) of section 128.01 of the Revised Code, common carrier services, and food, drink, transportation, entertainment, and cable television services and, for purposes of section 2913.04 of the Revised Code, include cable services as defined in that section.
It is not clear that that includes web subscription services, but since the provision says "include" it might, even though they are not explicitly mentioned.
In any case, such conduct is almost certainly a breech of the subscription agreement or TOS between the service provider and the subscriber (assuming, as the question seems to, that the person logging in is a valid subscriber). This would depend on the wording of that agreement, but pretty much all such agreements prohibit using services in excess of those explicitly authorized.
If the service provider became aware that a subscriber had abused the service by getting more free downloads or accesses than the authorized one per week, the provider could sue for breech of contract. Or the provider could simply terminate the subscription.
The provider might also report this to law enforcement, but in many places, this sort of theft is not a high priority for law enforcement unless it is being carried out on a quite large sale. If unlawfully downloaded videos are being resold, that might be more likely to trigger law enforcement action.
Since the abusive subscriber is perhaps copying videos in excess of what is permitted by the license that is part of the subscription, s/he is also committing copyright infringement, and the provide could also proceed with an infringement suit. But in many cases a provider will not think such a suit worth bringing.
The provider could, of course, take steps to fix the technical loophole that makes the unauthorized downloads possible (or try to fix it), but that is not a legal issue, merely a practical one.