How can I become my spouse’s caregiver?

You know there should be some kind of compensation for a Spouse caring for a spouse. Sometimes a spouse has to quit a job or cut back on hours to care for a Spouse so this means losing income they may need.

It all really depends on your income. And a lot of people fall thru the cracks. You could try Medicaid. Like said, if he was in the service, call your County VA. Aid and Attendance can only be gotten by those who served during war-time. But it does not hurt to call and see what DH can get.

Technically everyone is a "caregiver" for their spouse.
I am guessing you are asking how to get paid as a caregiver.
If your husband is a Veteran the VA now has programs that will pay the spouse to be a caregiver. Contact the VA and ask to talk to a Social Worker or a Patient Advocate. (If he is not in the VA system contact your County Veterans Assistance Commission they can help determine his level of "service connected disability, if there is one or not, and start the process.)
Contact your local Area Agency on Aging and see if there are any programs you might qualify for.
You could also contact your local Senior Center, ask if they have a Social Worker and they may be aware of programs that may help you.

You become your husbands caregiver by doing everything required to take on his care. You don't need permission from anyone to take care of your spouse, you just do it.
I'm guessing though you're asking how you get paid to take care of your spouse, which is a totally different question. I personally don't know of any spouses that ever got paid for caring for their spouse nor did they ever even think about needing to be paid, as when we marry it's "for better or worse, in sickness and health, till death do us part," so we take on their care out of love and commitment.
I took care of my late husband for 24 1/2 years of our 26 year marriage and never once expected or wanted to be paid for it. In fact it never even crossed my mind. I did it because I loved him and knew that I could care for him better than anyone else.
Now I believe that if you're on Medicaid, they will allow someone a few hours a week to be paid, and perhaps if your spouse is a veteran, and qualifies, they offer something called Aide and Assist which may allow you to be paid something, but usually they will send an aide out to your house instead.

If you are asking how you can be PAID as an inhome caregiver to family, the short answer is that you likely cannot be.

You can become a caregiver. Are you wanting to get PAID for being a caregiver?

The hard reality of life is that spouse or family caregivers are not financially compensated for their role.

Please give us more details about your situation, so that we can properly address your concerns.