Is the fluid that exits the penis during movement considered sweat or another type of fluid, and is ablution invalidated if clothing touches the moisture inside the urethral opening?

If the fluid you find is irregular in its timing, then your ruling is that of one afflicted with perpetual ritual impurity ( سلس ): you perform ablution (wudu') after the entry of the prayer time and pray with that ablution as many prayers as you wish until the time exits. You must wash the site of impurity and bandage it, performing ablution for each prayer. This fluid cannot be considered sweat. You are permitted to follow the Maliki school in not using a cloth for protection, even though its use is recommended. The entry of a part of one's clothing into the urethral opening does not invalidate ablution.