Yes, you can get a Kenyan driving License but since you do not have a Kenyan ID your details won’t be accepted in NTSA TIMS (portal used for driving class registration). Therefore, you need to enroll to a driving school, then visit NTSA office to have your details manually updated in TIMS portal.
Alternatively, you can train in Uganda and you will still be allowed to drive in Kenya.
If you have a valid driving license, you can drive in any East African country without problems.