Like most things, I think it depends on both the area & the restaurant when it comes to more than one check/separate checks.
Here, it's pretty normal for a server to ask "Will this all be one check?" when he/she comes to a table w/ a group that's more than just a "single family".
Even at a nice, more upscale restaurant, it's not unusual for a large group to pay w/ separate checks. DH & I were once invited to a retirement celebration at a nicer restaurant w/ the understanding that everyone would be paying for their own meals. There were about 35-40 of us split into probably about 15-20 separate checks. The "host" paid for the guest of honor & the guest of honor's spouse.
When I was making reservations at a local restaurant for our older 2 kids & their friends for dinner before prom, I had to call group reservations. The lady w/ whom I spoke asked me if there would be more than one check. When I replied that, yes, they were planning on separate checks, but, if there needed to be only one, we could work it out, she assured me that more than one check was perfectly fine & she was just asking so she could let the server know ahead of time.
OP - I would say since you are staying a hotel (& not at your cousin's home), either your cousin plans to pay for your dinner or plans for both parties to pay separately.
Like another poster said, it's all in how they asked - "Let us take you to dinner when you're here!" or "We should all go out for dinner while you're here!"